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INDEFINITE INTEGRATION, , IIT JEE Syllabus, , 1. Integration of a Function, 2. Basic Theorems on Integration,  Standard Integrals , 4. Methods of Integration, 5. Integrates of Different Expression, , Total No. of questions in Indefinite Integration are:, Solved examples……….......………………..…50, Level # 1 …….……………………………….. 141, Level # 2 …….……………………………….…48, Level # 3 …….……………………………….…27, Level # 4 ……………………………………..…13, Total No. of questions…..………………….. 279, , ***, PHOTON, , INDEFINITE INTEGRATION, , 3

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1. Students are advised to solve the questions of exercises (Levels # 1, 2, 3, 4) in the, same sequence or as directed by the faculty members., 2. Level #3 is not for foundation course students, it will be discussed in fresher and, target courses., , Index : Preparing your own list of Important/Difficult Questions, Instruction to fill, (A) Write down the Question Number you are unable to solve in column A below, by Pen., (B) After discussing the Questions written in column A with faculties, strike off them in the, manner so that you can see at the time of Revision also, to solve these questions again., (C) Write down the Question Number you feel are important or good in the column B., , EXERCISE, NO., , COLUMN :A, , COLUMN :B, , Questions I am unable, to solve in first attempt, , Good/Important questions, , Level # 1, , Level # 2, , Level # 3, , Level # 4, , Advantages, 1. It is advised to the students that they should prepare a question bank for the revision as it is, very difficult to solve all the questions at the time of revision., 2. Using above index you can prepare and maintain the questions for your revision., PHOTON, , INDEFINITE INTEGRATION, , 4

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KEY CONCEPTS, The following integrals are directly obtained from the, derivatives of standard functions., , 1. Integration of a Function, Integration is a reverse process of differentiation. The, integral or primitive of a function f(x) with respect, to x is that function (x) whose derivative with respect, to x is the given function f(x). It is expressed, symbolically as -, ,  f (x) dx   (x), , i., ii., iii., , Thus, , iv., , , , d, [(x)] = f(x), f (x) dx   (x) , dx, , The process of finding the integral of a function is, called Integration and the given function is called, Integrand. Now, it is obvious that the operation of, integration is inverse operation of differentiation., Hence integral of a function is also named as, anti-derivative of that function., Further we observe that-, , d 2, , ( x )  2x , dx, , d 2, ( x  2)  2x  , dx, , d 2, ( x  k )  2x , , dx, , v., vi., vii., viii., ix., ,  2x dx = x, , 2, , + constant, , So we always add a constant to the integral of function,, which is called the constant of Integration. It is, generally denoted by c. Due to presence of this constant, such an integral is called an Indefinite integral., ,  0 . dx = c, 1 .dx = x + c,  k .dx = kx + c (k R), , , x n dx =, ,  x dx = log x + c,  e dx = e + c, 1, , e, , x, , , , x, , a x dx =, ,  k f(x) dx = k  f (x) dx., (ii) [f (x)  g(x) dx = f ( x ) dx ± g(x) dx, , , , (iii) d/dx(  f ( x ) dx ) = f(x), ,  tan x dx =log sec x +c = – log cos x + c, xi. cot x dx = log sin x + c, , xii. sec x dx = log(secx+tanx)+c, , x., , = – log(secx – tan x)+c, ,  x, = log tan    + c, 4 2, xiii., ,  d, , , ,  cos ec x dx = – log (cosec x + cot x) + c, x, = log (cosec x – cot x) + c = log tan   + c, 2, , (i), , (iv), , ax, + c = ax loga e + c, log e a, ,  sin x dx = – cos x + c,  cos x dx = sin x + c, , 2. Basic Theorems on Integration, If f(x), g(x) are two functions of a variable x and k is a, constant, then-, , x n 1, + c (n–1), n 1, ,  sec x tan x dx = sec x + c, xv.  cos ec x cot x dx = – cosec x + c, xvi.  sec x dx = tan x + c, xvii.  cos ec x dx = – cot x + c, xiv., , 2, ,   dx f (x)  dx = f(x), , 2, , 3. Standard Integrals, PHOTON, , INDEFINITE INTEGRATION, , 5

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 sinh x dx = cosh x + c, , xviii., , =, ,  cosh x dx = sinh x + c, xx. sech x dx = tanh x + c, , xxi. cos ech x dx = – coth x + c, , xxii. sech x tanh x dx = – sech x + c, , xxiii. cos ech x coth x = – cosech x + c, , , e ax, , xix., , , b, sin bx – tan –1    + c, a, , a 2  b2, e ax, , =, , 2, , xxxv., , 2, , , , xxiv., , 1, 1, x, dx =, tan–1   + c, 2, 2, a, x a, a, , xxv., , x, , xxvi., , a, , xxvii., , , , 2, , 1, 1, x–a, dx =, log , +c, 2, 2a, –a,  xa , , 1, 1,  xa , dx =, log , +c, 2, 2a, –x, x–a, , 2, , x, dx = sin–1   + c, a, a –x, 1, , 2, , 2, , x, = – cos–1   + c, a, xxviii., , x, dx = sinh–1   + c, a, x2  a2, , , , 1, , , , xxix., , 1, 2, , x –a, , 2, , = log (x +, xxx., , , , xxxi., , , , , =, , xxxiii., , , , xxxiv., , , , PHOTON, , x, 2, , to, , x, a2, . sin–1 + c, a, 2, , x, 2, , x2  a2 +, , x, a2, . sinh–1 + c, a, 2, , (a cos bx + b sin bx) + c, , a  b2, , b, , cos bx – tan –1  + c, a, , , e ax, a b, 2, , 2, , f, , [(x)] ' (x) dx, ,  f (t) dt., , Note:, In this method the integrand is broken into two factors, so that one factor can be expressed in terms of the, function whose differential coefficient is the second, factor., (ii) When integrand is the product of two factors, such that one is the derivative of the other i.e., , x 2 – a 2 dx, , x, a2, . cosh–1 + c, a, 2, 1, x, 1, dx = sec–1 + c, 2, 2, a, a, x x –a, x, 2, , e ax, 2, , Here we put  (x) = t so that '(x) dx = dt, and in that case the integrand is reduced, , x2 – a2 ) + c, , a2 – x2 +, , cos bx dx, , When integration can not be reduced into some, standard form then integration is performed using, following methods(i) Integration by substitutions, (ii) Integration by parts, (iii) Integration of rational functions, (iv) Integration of irrational functions, (v) Integration of trigonometric functions, 4.1 INTEGRATION BY SUBSTITUTION:, Generally we apply this method in the following two, cases., (i) When Integrand is a function of function i.e., , x 2  a 2 dx, =, , xxxii., , dx = cosh–1, , ax, , 4. Methods of Integration, , x,  +c, a, , a 2 – x 2 dx, =, , =, , x2  a2 ) + c, , = log (x +, , e, =, , (a sin bx – b cos bx) + c, , a 2  b2, , x2 – a2 –, , e ax sin bx dx, INDEFINITE INTEGRATION, , I=, ,  f (x) f(x) dx., , In this case we put f(x) = t and convert it into a, standard integral., (iii) Integral of a function of the form f (ax + b)., Here we put ax + b = t and convert it, into, standard, integral., Obviously, if, ,  f (x) dx = (x), then,  f (ax  b) dx, , 1, (ax + b), a, (iv) Standard form of Integrals:, =, , 6

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f ( x ), , (a), ,  f (x ) dx = log [f(x)] + c, , (b), , , , [f (x)]n f (x) dx =, , x–a, or, x, , [f(x)] n 1, +c, n 1, , (provided n  – 1), f ( x ), dx = 2 f ( x ) + c, (c), f (x), , x ( x – a ) or, , (vii), , , , (v) Integral of the form, dx, a sin x  b cos x, , (viii), , , , , , 4.2 INTEGRATION BY PARTS :, 4.2.1, , Integrand form, , Substitution, , a 2 – x 2 or, , x = a sin  or, x = a cos , , (ii), , x 2  a 2 or, , x = a tan  or, , 1, , x = a cot  or x = a sinh , , x  a2, 2, , (iii), , x = a sec  or, , x 2 – a 2 or, , 1, , Note :, (i) From the first letter of the words inverse circular,, logarithmic,, Algebraic,, Trigonometric,, Exponential functions, we get a word ILATE., Therefore first arrange the functions in the order, according to letters of this word and then, integrate by parts., (ii) For the integration of Logarithmic or Inverse, trigonometric functions alone, take unity (1) as, the second function., 4.2.2, , x (a  x ) or, , or, , (v), , x, or, a–x, , 1, x (a  x ), , PHOTON, , or, , 4.2.3, , x, , f(x) + c, , If the integral is of the form, , [x f (x)  f(x)]dx, , then by breaking this integral into two, integrals integrate one integral by parts and, keep other integral as it is, by doing so, we get, ,  [x f (x) + f(x) ]dx = x f (x) + c, 4.3 Integration of Rational functions:, 4.3.1, When denominator can be factorized (using, partial fraction) :, , x (a – x ), x, x–a, , x, , x = a sin2 , , 1, , or, ,  e [f (x)  f (x)] dx = e, , x = a tan2 , , a–x, x, , x (a – x ), , or, , (vi), , ax, x, , , , If the integral is of the form e x [f(x) + f'(x)]dx,, then by breaking this integral into two, integrals, integrate one integral by parts and, keep other integral as it is, By doing so, we get-, , x = acosec orx = a cosh , , x, or, ax, ,  (u.v) dx = u   v dx  –   dx  .   v dx dx., , i.e. Integral of the product of two functions, = first function × integral of second function, –  [(derivative of first) × ( Integral of second)], , x2 – a2, (iv), , If u and v are two functions of x, then,  du , , (vi) Standard Substitutions : Following standard, substitutions will be useful-, , a2 – x2, , x = cos2  + sin2 , , ( > ), , 1, = log tan (x/2 + /2) + c, r, 1, log tan (x/2 + 1/2 tan–1 b/a) + c, 2, a  b2, , 1, , x–, or, –x, ( x – ) ( – x ), , putting a = r cos  and b = r sin . we get, dx, 1, I=, =, cos ec (x  ) dx ., r sin(x  , r, , (i), , x = a cos 2, , ax, a–x, , , , =, , a–x, or, ax, , x = a sec2 , INDEFINITE INTEGRATION, , 7

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f (x), , where both f(x), g(x ), and g(x) are polynomials. If degree of f(x) is greater, than degree of g(x) then first divide f(x) by g(x) till, the degree of the remainder becomes less than the, degree of g(x). Let Q(x) is the quotient and R(x), the, remainder then, , In this case integral may be in the form, , Let the integrand is of the form, , (i), , (ii) For repeated linear factors in the denominator,, writeA, B, 1, =, +, (x, –, a), (x – a) ( x – b), (x – a)2, ,  I = P/2a, , 3, , +, , 1, Ax  B, C, =, +, ax 2  bx  c x – d, (ax 2  bx  c)(x – d), , 4.3.3, , , , a–b, , , (x, , a)(x, , b), , , , , , PHOTON, , ax  bx  c, , ,, , , , ax 2  bx  c dx, , ax2 + bx + c = (x + )2 + , Also for integrals of the form, , , , x, then, ( x  a )( x  b), , px  q, ax  bx  c, 2, , dx,, ,  (px  q), , ax 2  bx  c dx., , First we express px + q in the form, , d, , px + q = A  (ax 2  bx  c) + B and then proceed, dx, , , as usual with standard form., , 1  b( x  a ) – a ( x  b) , , , b – a  ( x  a ) ( x  b) , , a , 1  b, –, b – a  x  b x  a , , dx, 2, , then we integrate it by expressing, , x, 1  (b – a ) x , =, , , ( x  a )( x  b) b – a  ( x  a )( x  b) , , =, , Integration of rational functions containing, only even powers of x., , If anyone term in Nr or Dr is irrational then it is made, rational by suitable substitution. Also if integral is of, the form-, , 1 ,  1, x  b – x a, , , , (ii) If integrand is of the form, , dx,  bx  c, , 4.4 Integration of irrational functions :, ,  ( x  a ) – ( x  b) , 1, =, , , (a – b)  (x  a)(x  b) , , 1, (a – b), , 2, , To find integral of such functions, first we divide, numerator and denominator by x2, then express, numerator as d(x ± 1/x) and denominator as a, function of (x ± 1/x)., , (i) If integrand is of the form, , 1, 1, =, (x  a)(x  b) (a – b), ,  ax, , Now we can integrate it easily., , Note :, 1, then use, (x  a)(x  b), the following method for obtaining partial, fractions-, , 2ax  b, dx, 2,  bx  c, ,  ax, , pb , , + q – , 2a , , , D, C, +, 3, (x, – b), (x – a), , (iii) For every non repeated quadratic factor in the, denominator, write, , 4.3.2, , (px  q), dx, 2,  bx  c, ,  ax, , b 2 – 4ac, 4a 2, Now the integrand so obtained can be evaluated, easily by using standard formulas., (ii) Here suppose that px + q = A [diff. coefficient, of (ax2 + bx + c) ] + B, = A (2ax + b) + B ...(1), Now comparing coefficient of x and constant, terms., we get A = p/2a, B = q – (pb/2a), , 1, A, B, =, +, (x – a)(x – b) x – a x – b, , =, , dx, , (ii),  bx  c, , x2 + (b/a) x + c/a = (x + b/2a)2 –, , Now in R(x)/g(x), factorize g(x) and then write partial, fractions in the following manner(i) For every non repeated linear factor in the, denominator, write, , =, , 2, , Method:, (i) Here taking coefficient of x2 common from, denominator, write -, , R (x), f (x), = Q(x) +, g(x ), g(x ), , Here, ,  ax, , 4.5 Integration of Trigonometric functions :, Here we shall study the methods for evaluation of, following types of integrals., , When denominator can not be factorised:, INDEFINITE INTEGRATION, , 8

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I., , (i), ,  a  b sin, , (ii), ,  a  b cos, , (iii), ,  a cos, , (iv), , , , dx, , 2, , 5. Some Integrates of Different Expression, of ex, , x, , dx, , 2, , 2, , aex, dx, b  cex, 1, (ii), dx, 1  ex, (i), , x, , dx, x  b sin x cos x  c sin 2 x, , , , , , (iii), , , , , , , , , 1, cos ec ( x  ) dx ., r, 1, = log tan (x/2 + /2) + c, r, 1, b, x 1, =, log tan   tan –1  + c, 2, 2, 2, 2, a, , a b, p sin x  q cos x, ,  a sin x  b cos x dx, p sin x,  a sin x  b cos x dx, q cos x,  a sin x  b cos x dx, , 1– e, 1, , x, , [Multiply and divide, , dx, , by e–x and put e–x=t], (iv), , e, , x, , 1, dx, – e–x, , [Multiply and divide, by ex], , (v), (vi), , , , ex – e– x, dx, ex  e–x, ,  f ' (x), , form , ,  f (x), , , , , ex  1, dx, ex – 1, , [Multiply and divide bye–x/2], ,  ex – e–x, (vii)  x, –x, e –e, , , , 2, , ,  dx, , , , [Integrand = tanh2 x], , 2, ,  e2x  1 ,  dx, (viii)  2 x, ,  e –1, 1, (ix), dx, x, (e  e – x ) 2, , , , , , (x), ,  (e, , (xi), , , , , , III., , [Multiply and divide, by e–x and put e–x=t], , dx, (a sin x  b cos x) 2, , Method :, Divide numerator and Denominator by cos2 x in all, such type of integrals and then put tan x = t., dx, II. (i), a  b cos x, dx, (ii), a  b sin x, dx, (iii), a cos x  b sin x, dx, (iv), a sin x  b cos x  c, Method :, In such types of integrals we use following formulae, for sin x and cos x in terms of tan (x/2)., x, x, 1 – tan 2  , 2 tan ,  2  , cos x =, 2, sin x =, 2 x , 2 x , 1  tan  , 1  tan  , 2, 2, and then take tan(x/2) = t and integrate another method, for evaluation of integral, (iii) put a = r cos , b = r sin , then, 1, dx, I=, r sin(x  ), =, , [put ex = t], , 1, dx, – e –x )2, 1, dx, x, (1  e )(1 – e – x ), x, , [Integrand = coth2 x], [Integrand = 1/4 sech2 x], [Integrand=1/4cosech2 x], [Multiply and divide by ex, and put ex = t], , (xii), , , , 1, , (xiii), , , , (xiv), , , , (xv), , , , 1 – ex, 1, , dx, , 1  ex, 1, , ex – 1, 1, 2e x – 1, , [Multiply and divide bye–x/2], , dx, , [Multiply and divide by e–x/2], , dx, , [Multiply & divide by e–x/2], , dx, , [Multiply and divide by, 2 e–x/2], , (xvi), , For their integration, we first express Nr. as followsNr = A (Dr) + B (derivative of Dr.), Then integral = Ax + B log (Dr) + C, , , , 1 – e x dx, , [Integrand, = (1 – ex)/ 1 – e x ], , (xvii), , , , 1  e x dx, , [Integrand, =(1 + ex) / 1 e x ], , PHOTON, , INDEFINITE INTEGRATION, , 9

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(xviii), , , , e x – 1 dx, , [Integrand, = (ex – 1) / e x – 1 ], , (xix), , , , ex  a, dx, ex – a, , [Integrand, = (ex + a)/ e 2 x – a 2, , PHOTON, , INDEFINITE INTEGRATION, , 10

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SOLVED EXAMPLE, ,  sin, , Ex.1, , Sol., , 2, , Sol., , (x / 2) dx equals-, , (A), , 1, 1, (x + sin x) + c (B) (x + cos x) + c, 2, 2, , (C), , 1, (x – sin x) + c (D) None of these, 2, , Here I =, =, , 2, , , , Ans.[C], , (B) – cot x – x + c, (D) None of these, Ex.5, , (cosec2 x – 1) dx, , = – cot x – x + c, , Ex.3, , [(a – b)3 = (a3 – 3a2b + 3ab2 – b3)], , , , , , Sol., , (B) 7x + 5 log x +c, (D) None of these, , 3, , =, , x, , dx – 3 xdx +3, , =, , x 31, x 31, x11, – 3., + 3log x –, +c,  3 1, 3 1, 11, , =, , 1, x4 3 2, – x + 3 log x + 2 + c Ans.[B], 2, 4, 2x, , Sol., , 7, 1, = 5 1 dx + 7, dx, x, x, = 5x + 7 log x + c, Ans.[C], , , , 3, ,  3x , , , , , , 1, ,   x  x , , , , (B) 6 tan–1 x +, , 10 x, +c, log e 10, , (C) 3 tan–1 x +, , 10 x, +c, log e 10, , , , 6, ,   1  x 2  10, =6, , x, , ,  dx, , , 1, ,  1  x 2 dx + 10, , 3, , x, , dx, , 10 x, +C, log e 10, , Ans.[B], , dx, (x > 0) equalsEx.6, , 3, , 1, 3, x, (A), – x2 + 3 log x + 2 + c, 2, 3, 2x, 1, x4 3 2, (B), – x + 3 log x + 2 + c, 3 2, 2x, , (C), , 1, ,  6, ,  10 x  dx is The value of  , 2,  1 x, , , = 6 tan–1 x +, Ex.4, , 1, ,  x dx –  x 3 dx, , (D) None of these, , 5x  7,  5x 7 ,   dx, dx =  , x,  x x, , =  5 dx + , , 3 1 , ,  dx, x x3 , ,   x, , (A) 6 tan–1 x + 10x loge 10 + c, , Ans. [B], , 5x  7, dx equalsx, (A) 5x + 7 log x, (C) 5x + 7 log x + c, , , , =, , x dx equals -, , (A) – sec x + x + c, (C) – sin x + x + c, Sol., , 1, 1, 1 , , =   x 3  3x 2 .  3x. 2  3  dx, x, x, x , , , 1  cos x, dx, 2, , 1, (x – sin x) + c, 2, ,  cot, , Ex.2, , , , , , 3, , 1, ,  x   dx, x, , , Sol., ,  (tan x  cot x), , 2, , dx is equal to-, , (A) tan x – cot x + c, , (B) tanx + cotx + c, , (C) cot x – tan x + c, , (D) None of these, , , , I = (tan 2 x  cot 2 x  2) dx, =  (sec2 x  cosec 2 x ) dx, , 1, x4, + 3 log x + 2 + c, 4, 2x, , = tan x – cot x + c, , Ans. [A], , (D) None of these, , PHOTON, , INDEFINITE INTEGRATION, , 10

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 sin 2x sin 3x dx equals-, , Ex.7, , , , (A), , 1, (sin x – sin 5 x) + c, 2, , (B), , 1, (sin x – sin 5x) + c, 10, , =, , 1, tan t + c, a, , (C), , 1, (5 sin x – sin 5x) + c, 10, , =, , 1, tan (ax + b) + c, a, , 1, 2, ,  [cos (x)  cos 5x] dx, , =, , 1, 2, , sin 5x , , sin x  5  + c, , , , 1, [5 sin x – sin 5x] + c, 10, , Ex.10, , Ans. [C], , Sol., , , ,  x 1 ,  x 1, (C) x + log ,  + c (D) x + log , + c,  x 1,  x 1 , , , =, , x 11, x 2 1, , , Ex.11, 1, , , ,  1  x 2  1  dx, 1,  x 1 , log ,  +c, 2,  x 1, , Sol., , 1, dx, log x, , 1, dx = dt, x, , 1, 1, ., dx =, x log x, , 1, , t, , dt, , 1, ,  t dt = log t + c = log (log x) + c, Ans.[D], ,  sec, , 2, , Ans.[A], , = sin (tan x) + c, , (B), , 2, , Ans.[B], ,  tann x sec2 x dx equalstan n 1 x, +c, n 1, (D) None of these, , tan n 1 x, +c, n 1, (C) tann+1 x + c, (A), , 1, tan x + c, 2, , 1, (C) tan (ax + b) + c (D) None of these, a, ,  sec, , (A) sin (cos x) + c, (B) sin (tan x) + c, (C) cosec (tan x) + c (D) None of these, Let tan x = t, then sec2 x dx = dt, , , , (ax + b) dx equals-, , (A) tan (ax + b) + c, ,  sec2x cos (tan x) dx equals-, ,  I = cos t dt = sin t + c, , Ex.12, , Sol., , (ax + b) dx, putting ax + b = t,, , dt, adx + 0 = dt or dx =, a, PHOTON, , ., , (B) log (log x + x) + c, (D) log (log x) + c, , (putting the value of t = log x), , dx, , x 1, = x + log, +c, x 1, , Sol., , Ans.[C], , 2, , =x+, , Ex.9, , 1, , 1, ,  x log x dx =  x, , , , x 1, x 1, (A) x + log, + c (B) x + log, +c, x 1, x 1, , Sol., ,  sec2t dt, , 1, , put log x = t,, ,  x 2 1 dx equals-, , dt, a, ,  x log x dx is equal to(A) log (x log x) + c, (C) log x + c, , x2, , Ex.8, , 1, a, , (Putting the value of t), , I=, , =, , = sec2 t, =, , (D) None of these, , Sol., , , ,  sec2 (ax + b) dx, ,  tann x sec2 x dx, sec2 x dx = dt, , putting tan x = t,, , , , , , tann x sec2 x dx = tn dt =, =, , INDEFINITE INTEGRATION, , (B), , (tan x ) n 1, +c, n 1, , tan n 1, +c, n 1, Ans.[B], 11

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sin 2 x, ,  1  cos 4 x dx is equal to-, , Ex.13, , Ex.16, , (A) cos–1 (cos2 x) + c (B) sin–1(cos2 x) + c, Sol., , (C) cot–1 (cos2 x) + c (D) None of these, Here differential coefficient of, cos2 x is – sin 2x, Let cos2 x = t, , Sol., , tan x, ,  sin x cos x dx equals(A) 2 sec x + c, , (B) 2 tan x + c, , (C) 2 / tan x + c, , (D) 2 / sec x + c, ,  2 cos x (– sin x) dx = dt, =, , or sin 2x dx = – dt,  , , 1  cos x, 4, , be x, , , , Ex.14, , sin 2 x, , a  be x, 2, b, , (A), , tan x, sec2 x dx, tan x, , I =, , sec2 x, , , , dt, ,  1 t2, , dx =, , Ex.17, , =, , cot–1, , t+c, , =, , cot–1, , (cos2 x) + c, , Ans.[C], , dx equals-, , a  be x + c, , (B), , 1, ., b, , a  be x +c, , Sol., , dx = 2 tan x + c, , Ans. [B], , tan x, ,  sin x. cos3 x dx is equal to5, , (A), , sin 6 x sin 8 x, cos6 x cos8 x, + c (B), +c, , , 6, 6, 8, 8, , (C), , cos6 x sin 8 x, + c (D) None of these, , 6, 8, ,  sin, , 5, , x . cos3 x dx, , Assumed that sin x = t, a  be x + c, , (C) 2, , , , Sol., , be, ,  cos x dx = dt, , (D) None of these, , , , x, , a  be x, , bex dx = dt, , , , , be x, a  be, , x, , dx =, , , , dt, t, , =2 t +c, , = 2 a  be x + c, , , , Ex.15, , Ex.18, , x, (C) 2 log sec   + c (D) None of these, 2, , I =, , , , 2 cos2 ( x / 2), , =, , =, , 2 sin 2 ( x / 2), , =, , sin 6 x sin 8 x, +c, , 6, 8, ,  1  x 6 dx is equal to(C), , Sol., , Ex.19, dx, , Ans.[A], , x2, , (B) tan–1x2 + c, , 1, tan–1x3 + c, 3, , (D) 3 tan–1x3 + c, , Put x3 = t  x2 dx =, I=, , , , 1, 3, , dt, ,  1 t2 =, , 1, dt, 3, , 1, tan-1 x3 + c, 3, , Ans. [C], , 1 x, dx equals1 x, , (A) sin–1 x + 1 x 2 + c, , x, ,  cot  2  dx, , x, = 2 log sin   + c, 2, PHOTON, , t 6 t8,  +c, 6 8, , (A) tan–1x3 + c, , x, x, (A) log cos   + c (B) 2log sin   + c, 2, 2, , Sol., , =, , Ans.[C], , 1  cos x, dx equals1  cos x, , 1  cos x, dx, 1  cos x, , , , = t5(1 – t2) dt = (t5 – t7) dt, , dx, putting a + bex = t, , (B) sin–1 x + x 2  1 + c, Ans.[B], , (C) sin–1 x – 1 x 2 + c, (D) sin–1 x – x 2  1 + c, , INDEFINITE INTEGRATION, , 12

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Sol, , Ex.20, , 1 x, dx, 1 x, , I =, , 1, –, 2, 2, 1 x, dx, , =, , , , =, , sin–1, , Ex.22, , , , 2x dx, ,  sin (log x) dx equalsx, , (A), , 2, , x – 1 x + c, , Ans. [C], , e, (B) x log   + c, x, , (D) None of these, Sol., ,  sin (log x) dx, assumed that x = et,  dx = et dt, , (D) x log (ex) + c, , =, ,  x tan, , Ex.21, , 1 2, (x + 1) tan–1 x – x + c, 2, , (B), , 1 2, (x + 1) tan–1 x + x + c, 2, , Ex.23, Ans. [C], , Sol., , ex, ( x  1), , =, =, , , )+c, 4, , Ans. [B], , ( x  1) 2, , ex, +c, x 1, , +c, , (B), , +c, , (D) None of these, ,  (x  1)  1, I = ex , dx, 2 ,  (x  1) , , , , =, , 1 2, 1, (x – 1) tan–1 x – x + c, 2, 2, Integrating by parts taking x as second part, , 2, , ex, , e, , (D), , x, , , 1,  1 , dx, ,  x  1 ( x  1) 2 , , , , = ex f(x) + c, =, Ex. 24, , 1 , 1, 1 , = x2 tan–1 x – 1 ,  dx, 2, 2  1 x2 , , sin (log x –, ,  (x  1) 2 dx is equal to-, , (C), , 1, x2, x2, I=, tan–1 x – , ., dx, 2, 1 x2 2, , x, , xe x, , (A), , 1, 1, (C) (x2 + 1) tan–1 x – x + c, 2, 2, , Sol., , sin(t – tan–11) + c, , 11, , 2, , x is equal to-, , (A), , et, , =, , 1, .x dx = (x logx – x) + c, x, , x, = x (log x –1) + c = log   +c, e, ,  sin t.et.dt, ,   sin (log x) dx, ,  d (log x ) , = (log x).x –  ,  . x dx,  dx , , 1, , , )+c, 4, , 2, , [Integrating by parts, taking log x as first part and, 1 as second part], , , , cos (log x –, , x, , (C), ,  log x dx =  log x.1dx, , = x log x –, , , )+c, 4, , 2, , =, , Sol., , sin (log x –, , x, , (B), , The primitive of log x will be-, , x, (C) x log   + c, e, , , )+c, 8, , 2, , 1 x2, , (A) x log (e + x) + c, , sin (log x +, , x, , ex, +c, x 1, 3, , Ans. [B], , (log x ) 2 dx equals-, , (A), , 1 2 –1, 1, 1, x tan x – x + tan–1 x + c, 2, 2, 2, , 1 4, x [8 (log x)2 – 4 log x + 1] + c, 32, , (B), , 1 2, 1, (x + 1) tan–1 x – x + c, 2, 2, , 1 4, x [8 (log x)2 – 4 log x – 1] + c, 32, , (C), , 1 4, x [8 (log x)2 + 4 log x + 1] + c, 32, , Ans. [C], , (D) None of these, PHOTON, , INDEFINITE INTEGRATION, , 13

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Sol., , Integrating by parts taking x3 as second part, , 1, 1, I = x4(log x)2 –, 4, 2, , x, , 3, , Sol., , Ex. 28, ,  sec2 x  tan 2 x , +c, = x sec x – log ,  sec x  tan x , , , , Sol., , e, , x, , , 1, , 2, , 2, , , , 3, , ex, ( x  1) 2, , Ans.[C], , 1, [tan  sec – log (tan+ sec)] + c, 2, (D) None of these, Sol., , I=, , Ans. [C], =, , , , tan 2   1 sec2  d, , =, , , , t 2 1 dt, where t = tan , , =, , x, tan x + c, 2, , (D) x tan x + c, Ex.29, ,  sec. sec2. d, , , , t, 2, , t2 1 +, , 1, log  t  t 2  1  + c, , , 2, , 1, [tan sec + log (tan + sec)] + c, 2, Ans. [A], , cos x  x sin x, dx is equal tox ( x  cos x ), , , 1, 2, ,  x sec (x / 2) dx +  tan (x / 2) dx, , x, , , (B) log , + c, x, , cos, x, , , , = x tan (x/2) –  tan (x / 2) dx +  tan (x / 2) dx, ,  x  cos x , (C) log , + c,  x  cos x , , dx, , (A) log {x (x + cos x)}+ c, , 2, , x, = x tan   + c., 2, , +c, ,  sec3d is equal to-, , x  2 sin ( x / 2) cos(x / 2), , =, , PHOTON, , , , =, , 2 cos 2 ( x / 2), , +c, , (C), , x  sin x, , I=, , ( x  1) 2, , 1, [tansec+ log (tan+ sec)] + c, 2, 1, (B) tan  sec + log (tan+ sec) + c, 2, ,  1  cos x dx equals-, , x, (C) x tan   + c, 2, , ex, , (A), , sec x  tan x , , = x sec x – log (sec x  tan x ), + c, sec x  tan x , , , (B), , (D) –, , +c, ,  I = ex f(x) =, , (Integrating by parts, taking x as first, function), = x sec x – log (sec x + tan x) + c, , x, x, tan   + c, 2, 2, , ( x  1) 2, , ex, +c, x 1, , , , , , (A), , ex, , (B), , = e x {f (x)  f (x)} dx, , = (x. sec x) – (1.sec x) dx, , Ex.26, , ex, +c, x 1, ,  x 1 2 , I = ex , dx, 3 ,  ( x  1) , =, ,  x .(sec x tanx) dx, , = x sec x + log (sec x – tan x) + c, , dx equals-, , ,  dx,  ( x  1), ( x  1) , , Thus the given integral is of the form, ,  x sec x tan x dx is-, , (A) x sec x + log (sec x + tan x) + c, (B) x sec x – log (sec x – tan x) + c, (C) x sec x + log (sec x – tan x ) + c, (D) None of the above, Sol., , ( x  1) 3, , (C), , Ans. [A], The value of, , x 1, , x, , (A) –, , 1 4, x [8 (logx)2– 4 log x +1] + c, 32, , =, , e, , log x dx, , 1, 1 4, 1 1, , x (log x)2 –  x 4 log x  x 4  + c, 16 , 4, 2 4, , =, , Ex.25, , Ex.27, , Ans.[C], INDEFINITE INTEGRATION, , (D) None of these, 14

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Sol., , I =, , ( x  cos x )  x  x sin x, dx, x ( x  cos x ), , The above method is used to obtain the value of, constant corresponding to non repeated linear, factor in the Den r., , 1  sin x, 1, =, dx – , dx, x  cos x, x, , , , 2, 5 , , , Now I = 1 ,  dx,  x 1 x  2 , , = log x – log (x + cos x) + c, , = x – 2 log (x – 1) + 5 log (x – 2) + c, , x, , , = log , + c, x, , cos, x, , , , , , Ex.30, , Ans. [B], ,  ( x  2) 5 , = x + log , +c, 2 ,  ( x  1) , , sec x 1 dx is equal to-, , (A) 2 sin–1 ( 2 cos x / 2) + c, Ex.32, , x 2 dx, ,  (x 2  a 2 ) (x 2  b 2 ) is-, , The value of, , (B) –2 sinh–1 ( 2 cos x / 2) + c, (C) –2 cosh–1 ( 2 cos x / 2) + c, , (A), , (D) None of these, Sol., , I =, =, , , , = –2, , 1  cos x, dx, cos x, , 2 sin x / 2, 2 cos2 x / 2 1, , dt, , , , t 2 1, , Sol., , I=, , x 1, ,  ( x  2) 5 , (A) log , +c, 2 ,  ( x  1) , , =, , 2, , 1, b2  a 2, , =, , 1, b a, 2, , 2, ,  b2, a2 , , , , 2, y  a 2 ,  y  b, , , , b2, a2, .  2, dx, , dx, 2, 2, 2, x a,  x  b, , , , , 1, , 1 x, 1, b tan b  a tan, b2  a 2 , , , , x, +c, a , , Ans.[A], ,  ( x  2) 5 , (B) x + log , +c, 2 ,  ( x  1) , , Ex.33, ,  ( x  1) 5 , (C) x + log , +c, 5,  ( x  2) , , dx, ,  3x 2  2x  1 equals(A), , (D) None of these, Here since the highest powers of x in Numr and, Denr are equal and coefficients of x2 are also, equal, therefore, , (B), , (C), , A, B, x 2 1, 1+, +, x 1, x2, ( x  1) ( x  2), , On solving we get A = – 2, B = 5, , PHOTON, , , 1 x, 1 x , b tan b  a tan a  + c, , , , (y  a ) (y  b ), ,  (x 1) (x  2) dx equals-, , x, +c, a , , (D) None of these, Putting x2 = y in integrand, we obtain, , Ans. [C], , 2, , Thus, , 2, , 2, , = – 2 cosh–1 + ( 2 cos x/2) +c, , Sol., , b a, 2, , y, , = –2 cosh–1 t + c, , Ex.31, , , 1 x, 1 x , b tan b  a tan a  + c, , , , 1, , (C), , where t = 2 cos x/2, , b a, , 2, , 1, , 1 x, 1, a tan b  b tan, b2  a 2 , , (B), , dx, , 1, 2, , Ans.[B], ,  3x  1 ,  + c, tan–1 , 2,  2 , , 1, ,  3x  1 ,  + c, sin–1 , 2,  2 , , 1, ,  3x  1 ,  + c, cot–1 , 2,  2 , , 1, , (D) None of these, Sol., , 2, 5, x 2 1, 1–, +, x 1 x  2, ( x  1) ( x  2), INDEFINITE INTEGRATION, , I=, , dx, 1, 3 x2  2 x  1, 3, 3, , 15

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=, , 1, 3, , dx, , , , Sol., , 2, , 1, 2, x   , 3, 9, , , , , 1,  x  , 1, 3, 3 c, = ×, tan–1 + , , 3, 2, 2 /3 , , , , ,  3x  1 ,  + c, tan–1 , 2,  2 , , 1, , =, , dx, , , , I=, , 37 , 5, x  , 4 , 2, , 2, ,  x 5/ 2 ,  + c, = sin–1 ,  37 / 2 ,  2x  5 ,  + c, = sin–1 ,  37 , , Ans.[A], Ex.36, , , , Ans. [A], , e 2 x 1 dx is equal to-, , (A), , e 2 x  1 + sec–1e2x + c, , 1, 9 2,  4x  1 , (A) (4x –1) 1  x  2x 2 +, sin–1 , + c, 8, 32,  3 , , (B), , e 2 x  1 – sec–1e2x + c, , (C), , e 2 x  1 – sec–1ex + c, , 1, 9 2,  4x  1 , (B) (4x +1) 1  x  2x 2 –, sin–1 , + c, 8, 32,  3 , , (D) None of these, , , , Ex.34, , 1  x  2x 2 dx equals-, , 1, 9 2,  4x  1 , (C) (4x –1) 1  x  2x 2 +, cos–1 , + c, 8, 32,  3 , , Sol., , =, , (D) None of these, Sol., , I=, , = 2, , 1  2 x,   x   dx, 2 , 2, , , , 2, , 2,  9 , 1  ,    x    dx, 4  , 16 , , , , 1 , 1, = 2  x  , 4, 2 , , Ex.37, , 2,  9 , 1  , , x, , , , , , 4  , 16 , , 4 , 9, 1 , +, sin–1   x   + c, 32, 4 , 3 , 1, 9 2,  4x  1 , = (4x –1) 1  x  2x 2 +, sin–1 , + c, 8, 32,  3 , , , , dx, 3  5x  x, , equals-, , Sol., , (A), , (B), , cos–1, , dx, , e 2x  1, , 1, 2, , , , 2e 2 x, , dx –, , e 2 x 1, ,  ex, , =, , e 2 x  1 – sec–1 ex + c, , , , ex  a, ex  a, , ex, e 2 x 1, , dx, Ans.[C], , dx is equal to-, ,  ex, (A) cosh–1 ,  a, , ,  x, ,  + sec–1  e,  a, , , , , , + c, , , ,  ex, (B) sinh–1 ,  a, , , ,  + sec–1, , , ,  ex, ,  a, , , , + c, , , ,  ex, (C) tanh–1 ,  a, , ,  x, ,  + cos–1  e,  a, , , , , , + c, , , , ex  a, , , , e 2x  a 2, , 2, ,  2x  5 ,  + c, ,  37 , , sin–1 , , e 2x  1, , (D) None of these, , Ans. [A], Ex.35, , , , =, , , , ex, e 2x  a 2, ,  ex, = cosh–1 ,  a, , ,  2x  5 , ,  + c,  37 , , dx, dx + a , , ex, e x e 2x  a 2, , ,  x,  + sec–1  e, ,  a, , , , , + c, , , , dx, , Ans.[A], , (C) sin–1 (2x + 5) + c, (D) None of these, PHOTON, , INDEFINITE INTEGRATION, , 16

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dx, ,  4 sin 2 x  4 sin x cos x  5 cos2 x is equal to-, , Ex.38, , Sol., , 1, , (C) 4 tan–1  tan x   + c, 2, , (D) None of these, After dividing by cos2 x to numerator and, denominator of integration, ,  ex  2 , t2, + c, = log ,  + c = log  x, ,  t3,  e 3, Ans. [B], Ex.41, , 2, , I=, =, =, , sec x dx, ,  4 tan 2 x  4 tan x  5, sec2 x dx, ,  (2 tan x  1) 2  4, , Sol., , 1,  2 tan x  1 , tan–1 , + c, 2, 2 .2, , , ,  1 x , ,   1  x , , Ex.39, , dx, ,  x, , (B) 2log ( x +1) +c, , (C) tan–1 x + c, , (D) None of these, , dx, ,  x, , I=, =, , 2 t dt, ,  t2  t, , dx is equal to-, , 4, +c, x 1, 4, (B) x – log (x +1) +, +c, x 1, 4, (C) x – 4 log (x +1) –, +c, x 1, 4, (D) x + log (x +1) –, +c, x 1, , , , Sol., , =, , ( x  1) 2, , , , =–, , ex, ,  e 2x  5e x  6, , Ex.40, , Sol., , 4, – 4 log (x + 1) + x + c, x 1, , dt, , I=, ,  9e x  4e x, , (A), , 19, 35, x+, log (9ex – 4e–x) + c, 36, 36, , 4e x  6e  x, , (C), ,  4, , 4, ,  1 dx, , 2, x  1 ,  ( x  1), Ans. [C], , 1, 1, x+, log (9ex – 4e–x) + c, 36, 36, , (D) None of these, Suppose 4ex + 6e–x = A (9ex – 4e–x) +, B (9ex + 4e–x), By comparing 4 = 9A + 9B ,, 6 = – 4A + 4B, 4, 3, ,–A+B=, 9, 2, , After solving A = –, ,  ex  2 ,  +c, (B) log  x,  e 3, , , , I=, ,  ex  2 , 1, + c, log  x,  e 3, 2, , , , Ans.[B], , dx is equal to-, ,  ex  3 ,  +c, (A) log  x, e 2, , , , (C), , x +1) + c, , 19, 35, x+, log (9ex – 4e–x) + c, 36, 36, , or A + B =, , equals-, , where t2 = x, ,  t 1 = 2 log (, , (B) –, , dx, , x, , =2, , (A) x – 4 log (x +1) +, , Ex.42, , equalsx, , (A) 2log ( x –1)+ c, , Ans. [B], , 2, , [2  ( x  1)] 2, , dt, , 1 ,  1, , = ,  dt,  t  2 t 3, , 1, 1, , (B) tan–1  tan x   + c, 2, 4, , , Sol., , dt, ,  t 2  5t  6 =  (1  2) (t  3), , I=, , 1, ,  tan x   + c, 2, , , (A) tan–1, , Put ex = t  ex dx = dt, , =–, , , , 19, 35, ,B=, 36, 36, ,  19 35  9e x  4e  x,   x, , x,  36 36  9e  4e, , ,  dx, , , , 19, 35, x+, log (9ex– 4e–x) + c, 36, 36, , Ans.[B], , (D) None of these, PHOTON, , INDEFINITE INTEGRATION, , 17

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, , Ex.43, , sin 1 x, , dx equals-, , 1 x, , Ex.45, , (A) 2[ x – 1 x sin–1 x ] + c, (B) 2[ x + 1 x sin, , –1, , , , (A), , 2 3, (x – 2) 1 x 3 + c, 9, , (B), , 2 3, (x + 2) 1 x 3 + c, 9, , (D) None of these, Sol., , Put 1 + x3 = t2  3x2 dx = 2 t dt, I=, , t sin t dt, , = 2[ x – 1 x sin–1 x ] + c, x a, xa, , dx equals-, ,  xa, (A) x 2  ax –2 ax  a 2 – a cosh–1 , , a, , , , +c, , , ,  xa, x  ax + ax  a – a cosh–1 + , , a, , 2, , (B), , 2, , , c, , , ,  xa , +c, x 2  ax – 2 ax  a 2 + a cosh–1 , , , a, , , (D) None of these, Sol., Let x = a tan2   dx = 2a tan  sec2 d, , Ex. 46, , (C), , , , I=, , a (tan   1).2a tan  sec , a sec , , =, , 2 1, , (1  x 3 ) 3 / 2  1  x 3  + c, , 3 3, , , =, , 2, 9, , =, , 2 3, (x – 2) 1 x 3 + c, 9, , = 2a, , , , d, , sec   1 tan  sec  d –sec ], , t 2 1 dt– 2a sec  + c [Where sec  = t], , 1, t 2, , t  1  cosh1 ( t ) – 2a, = 2a , 2, 2, , , =a, ,  xa, xa x, . – a cosh–1 , , a a, a, , , ax, +c, a, , , , , , , x 2  ax – 2, , Ex.47, , , +c, , , , Ans. [A], PHOTON, , x, , (1  x ) 2, , (1  x 2 ), 1, , x, , INDEFINITE INTEGRATION, , 1, , Ans. [A], , dx is equal to1, , +c, , (B) xe2 tan, , x, , (D) None of these, , +c, , I=, , 1, 2, ,  e {1  tan(t / 2)}, , =, , 1, 2, ,  e sec, , =, , 1 t, e (2 tan t/2), 2, , x, , +c, , t, , t, , 2, , 2, , dt, , t, t,  2 tan  dt, 2, 2, , 1, t, = xe2 tan x + c, 2, , Ans. [B], , If I =  cos 1 x dx and, J=, ,  xa, ax  a 2 – a cosh–1 , , a, , , 1, , = et tan, , – 2 ax  a 2 + c, =, , 1 x 3 (1+ x3 – 3) + c, , Putting the value of 2 tan –1 x = t, , = 2a [  tan  sec  d   sec  tan  d ], , , ,  t3 ,   t + c,  3, , , (C) 2xe2 tan, , 2, , 2, (t2 – 1) dt, 3, , 2, 3, , e 2 tan, , 2, , = 2a [, , 1 x3, , (A) xetan, , Sol., , 2, , , , (x2 dx) =, , =, Ans. [A], , , , x3, , , , = 2 [– t cos t + sin t] + c, , Ex.44, , dx equals-, , 1 x3, , (C) (x3 + 2) 1 x 3 + c, , t,  I=, . 2 sin t cos t dt, cos t, , =2, , , , x ]+c, , (C) [ x – 1 x sin–1 x ] + c, (D) None of these, Let x = sin2 t, then, dx = 2 sin t cos t dt, , Sol., , x5, , sin 1 x  cos1 x, ,  sin 1, , (A) x – 4I, (C) x –, , 4, I, , , x  cos1 x, , dx, then J equals-, , (B) x + I, (D), , , 4, 18

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Sol., , Here, , Ex. 50 If, , 2, J=, , ,  {sin, , x  cos, , 1, , x } dx, , [ sin–1, , x + cos–1, , 4, ,  dx –   cos, , =, , =x–, , 1, , x=, , , ], 2, , Sol., , x dx, , 4, I., , , Ans. [C], , Which value of the constant of integration will, , , , At x = 0, I = –, , If, , 1 1, + +c, 16 4, , x, , dx, , , 4, , Sol., , , , (B) –, , (C) , , (D), , 2, , Bx  C, A, + 2, x 1, x 1, , =, ,  2x + 3 = A(x2 +1) + (Bx + C) (x –1)...(1), Now putting x = 1, we get 5 = 2A A = 5/2, Equating coefficients of similar terms on both, sides of (1),, we get,, – B + C = 2, A – C = 3, , , 4, , , , dx, +, x 1, , , , , , 5, 1, x, 2, 2 dx, x2 1, , 2x, , 1, , 1, , =, , 5, 5, log(x–1) –, 2, 4, , =, , 1, 5, 5, log(x–1) – log(x2 +1) – tan–1x +c, 2, 4, 2, , Ans. [B], ,  1  sin x = tan  2  a  +c, then value of a is, , (A), , ( x  1) ( x  1), , 5, I=, 2, ,  I = 0  c = – 3/16, Ex.49, , 2x  3, , B = – 1/2 – 2 = – 5/2, , 1  cos 8x cos 2x , , , +c, 2 , 8, 2 , , =, , Let =, ,  C = 5/2 – 3 = – 1/2, , 1, (sin 8 x – sin 2x) dx, 2, , I=, , log[(x–1)5/2(x2+1)a], , 1, tan–1 x + k where k is any arbitrary constant,, 2, then a is equal to, (A) 5/4, (B) –5/3, (C) –5/6, (D) –5/4, , make the integral of sin 3x cos 5x zero at x = 0, (A) 0, (B) – 3/16, (C) –5/16, (D) 1/8, Sol., , =, , –, , 2 , , 1, x  dx,   2 cos,  2, , , =, , Ex.48, , 1, , 2x  3, ,  (x  1) (x 2  1) dx, ,  x 2 1 dx – 2  x 2 1 dx, , = log [(x–1)5/2 (x2 + 1)–5/4] –,  a = – 5/4., , 1, tan–1 x + c, 2, , Ans. [D], , , 2, , dx, ,  1  sin x, , I=, , dx, , =, ,  1  cos( / 2)  x, , =, , 1, 2, , 2, , , , x, ,  sec  4  2  dx, ,   x,  x, = – tan    + c = tan     + c,  4 2, 4 2, ,  = –, , PHOTON, , , 4, , Ans. [B], , INDEFINITE INTEGRATION, , 19

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LEVEL- 1, Question, based on, , Inequation, , , , Q.1, , Q.8, , 1  sin 2x dx equals-, , (A) sin x + cos x + c, (C) cos x – sin x + c, Q.2, , , , (B) sin x – cos x + c, (D) None of these, , 4  5 sin x, , dx equalscos 2 x, (A) 4 tan x – sec x + c, (B) 4 tan x + 5 sec x + c, (C) 9 tan x + c, (D) None of these, , Q.9, , Q.10, ,  (tan x + cot x) dx equals-, , Q.3, , (A) log (c tan x), (B) log (sin x + cosx) + c, (C) log (cx), (D) None of these, , Q.5, , Q.11, , e, , x2, (A), +c, 2, , x3, (B), +c, 3, , x4, (C), +c, 4, , (D) None of these, , 1  cos 2 x,  1  cos 2x dx equals(A) tan x + x + c, (B) tan x – x + c, (C) sin x – x + c, (D) sin x + x + c, ,  cos2x sin4x dx is equal to-, , Q.6, , (A) (a/b)x + 2x + c, , (B) (b/a)x + 2x + c, , (C) (a/b)x – 2x + c, , (D) None of these, , , , (B) tan x + cot x + c, , (C) cot x – tan x + c, , (D) None of these, , (A), , (C) 5n x c, PHOTON, , dx equals-, , (A), , 2 cos (x/2) + c, , (B), , 2 sin (x/2) + c, ,  sec x (tan x + sec x) dx equals-, , (D) None of these, Q.12, , The value of, , 1  sin 2x, , dx is-, , (B) x + c, , (C) cos x + c, , (D), , 1, (sinx + cos x), 2, ,  abx baxdx is] where a, b  R+, , (C), , x n 51, +C, n 5  1, , sin x  cos x, , , , (A) sin x + c, , (A), , Q.14, (B), , 1  cos x, , (C) tan x + sec x + c, , Q.13, , dx is equal to, , 5n x 1, +C, n x  1, , sin x, , , , (B) sec x – tan x+ c, , 1, (cos 6x + 3 cos 2x) + c, 12, (D) None of these, , 5, , equals-, , (A) tan x – cot x + c, , (C) –, , nx, , sin x cos 2 x, , (A) tan x – sec x + c, , 1, (A), (cos 6x + 3 cos 2x ) + c, 12, 1, (B), (cos 6x + 3 cos 2x) + c, 6, , Q.7, , dx, 2, , (D) –2 2 cos (x/2) + c, , e 5 loge x  e 4 loge x, e, , dx equals-, , a xbx, , (C) 2 2 cos (x/2) + c, ,  e3log x  e 2 log x dx equals-, , Q.4, , , , (a x  b x ) 2, , , , a bx b ax, n ( a b b a ), , +c, , a bx . b ax, n a b . n b a, , +c, , cos 2x  2 sin 2 x, cos2 x, , (B), , a bx . b ax, +c, n a . n b, , (D) None of these, , dx equals-, , (A) cot x + c, , (B) sec x + c, , (C) tan x + c, , (D) cosec x + c, , (D) None of these, INDEFINITE INTEGRATION, , 20

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, , Q.15, , sin x ,  1, cosx  2 ,  dx equals sin x cos3 x , , Q.21, , , , sin 4 x  cos4 x, sin 2 x cos2 x, , (A) sec x – cosec x + c, , (A) tan x + cot x – 2x + c, , (B) cosec x – sec x + c, , (B) tan x – cot x + 2x + c, , (C) sec x + cosec x + c, , (C) tan x – cot x – 2x + c, , (D) None of these, , (D) None of these, , , , Q.16, , sin 3 x  cos3 x, sin 2 x cos2 x, , Q.22, dx equals-, , (C) 2 [– sin x + x cos ] + c, , (C) sin x – cos x + c, , (D) – 2 [sin x + sin ] + c, , (D) None of these, ,  cos x, , Q.23, , cos 3x dx equals-, , (A), , 1, (sin 4x + 2 sin 2x) + c, 8, , (B), , 1, (sin 4x – 2 sin 2x) + c, 8, , (C), , 1, sin x sin 3x + c, 8, , Q.24, , , , Q.18, , 5x, , , , sin 2 x, (1  cos x ) 2, , (A) 2 tan x/2 + x + c, , (B) 2 tan x/2 – x + c, , (C) tan x/2 – x + c, , (D) None of these, , , , (2 / 5) x, (3 / 5) x, (A), +, +c, log e 2 / 5 log e 3 / 5, (B) loge (2x/5) + loge (3x/5) + c, , Q.25, , (D) None of these, sin 2 x, dx is1  cos x, , (A) x – sin x + c, , (B) x + sin x + c, , (C) – x – sin x + c, , (D) None of these, , , , Q.20, , x 1  x, , equals-, , (C), , 3, [(x +1)3/2 + x3/2] + c, 2, , (D), , 2, [(x +1)3/2 + x3/2] + c, 3, , , , (C) x + c, , , , dx, , (B) (x +1)3/2 – x3/2 + c, , dx equals-, , The value of, , dx equals-, , (A) (x +1)3/2 + x3/2 + c, , (D) None of these, 2 x  3x, , cos 2 x  cos 2, dx =, cos x  cos , , (B) 2 [sin x + sin ] + c, , (B) sec x + cosec x + c, , Q.17, , , , (A) 2 [sin x + x cos ] + c, , (A) sec x – cosec x + c, , Q.19, , dx equals-, , 1  tan x, dx equals1  tan x, , dx, 3x  4  3x  1, , equals-, , (A), , 2, [(3x + 4)3/2 – (3x + 1)3/2] + c, 27, , (B), , 2, [(3x + 4)3/2 + (3x +1)3/2 ]+ c, 27, , (C), , 2, [(3x + 4)3/2 – (3x +1)3/2] + c, 3, , (D) None of these, Q.26, , (A) log (cos x + sin x) + c, , a, , loga (sec2 x  tanx), , dx equals-, , (A) etan x + log sec x + c, , (B) log (cos x – sin x) + c, , (B) etan x + e logcos x + c, , (C) log (sin x – cos x) + c, , (C) tan x + log sec x + c, , (D) None of these, , (D) sec x + log cos x + c, PHOTON, , INDEFINITE INTEGRATION, , 21

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Question, based on, , Q.27, , Integration by substitution, , The value of, , dx, ,  (sec–1 x) x, , x2 –1, , Q.33, , Q.34, ,  (sec1 x ) 2, +c, 2, (D) None of these, ,  x 6 1 dx equals(B) tan–1 (x3) + c, (D) 3 tan–1 (x3/3) + c, , (A) log (x6 +1) + c, (C) 3 tan–1 (x3) + c, , Q.35, , Q.36, , Evaluate :  cot x cosec2 x dx., 1, cot2 x + c, 2, , (A) –, , 1, cot2 x + c, 2, , (B), , (C) –, , 1, cos2 x – c, 2, , (D) None of these, , Evaluate :, , (A), , 1, 2, , log(x  1  x 2 ), 1 x2, , dx, , Q.38, , (B) log(x  1  x 2 ) + c, , , , tan (log x ), isx, (A) log cos (log x) + c, (B) log sin (log x) + c, (C) log sec (log x) + c, (D) log cosec (log x) + c, , PHOTON, , , ,  tan x, , sec2 x, tan 2 x 1, , dx is equal to -, ,  tan3 x sec2 x dx1, tan4 x + c, 4, , (A), ,  (tan x)3 d (tan x) =, , (B), ,  (cos x)3 d (tan x) = 3 tan4 x + c, , (C), ,  (tan x)3 d (tan x) = – 4, , 1, , 1, , tan4 x + c, , , , (sec x cosecx), dx equalslog tan x, , , , x e 1  e x 1, xe  ex, , dx is equal to-, , (B) e log (xe– ex) + c, , 1 , log(x  1  x 2 ) + c, , 2 , , The value of, , a2  x2 + c, , (A) log (xe – ex) + c, (C) – log (xe– ex) + c, , (D) None of these, Q.32, , (D) a cos–1 (x/a) –, , (A) log log cot x + c (B) cot log x + c, (C) log (log tan x) + c (D) tan log x + c, , 2, , log(x  1  x 2 ) + c, , , 2, , (C), , a2  x2 + c, , (D) None of these, Q.37, , Q.31, , (C) a sin–1 (x/a) –, , (A) sec–1 (tan x) + c (B) sec (tan–1 x) + c, (C) cosec–1 (tan x) + c (D) None of these, , cos x,  1  sin x dx is equal to-, , (A) – log (1+ sin x) + c, (B) log (1+ sin x) + c, (C) log (1– sin x) – c, (D) log (1– sin x) + c, Q.30, , a2  x2 + c, , (B) cos–1 (x/a) – a 2  x 2 + c, , 3x 2, , Q.29, , (B) log (ex – e–x) + c, (D) tan–1 (e–x) + c, , ax, dx is equal toax, , , , (A) sin–1 (x/a) –, , (C), , Q.28, , equals-, , (A) log (ex + e–x) + c, (C) tan–1 (ex) + c, , is-, , (A) – log (sec–1 x) + c, (B) log (sec–1 x) + c, , dx, ,  e x  ex, , (D)(1/e)log(xe– ex) + c, Q.39, , INDEFINITE INTEGRATION, ,  sec4 x tan x. dx is equal totan 4 x, +c, 4, , (A), , sec4 x, +c, 4, , (B), , (C), , sec5 x, +c, 5, , (D) None of these, 22

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 x2 cos x3 dx is equal to-, , Q.40, , (A) 1/3 sin (x3) + c, (C) sin (x3) + c, Q.41, , Q.48, , (B) 3 sin (x3) + c, (D) –1/3 sin (x3) + c, , (C), , , , Q.42, , 1, sec2 + c, 2, , sin2, , x 2 tan 1 x 3, 1 x6, , Q.49, , (D) – cosec + c, , dx is equal to1, (B) (tan–1x)3+ c, 6, , 1, (C) (tan–1x)2 – c, 6, , 1, (D) (tan–1x3)2 + c, 6, , Q.50, , , , x2, , , , x3, , equals-, , (A) sec (3x – 5) + c, (C) tan (3x – 5) + c, , , , Q.46, , 2, , (A), , 1  tan 6 x, , tan–1(tan3x), , (B)1/3sec(3x–5)+ c, (D) None of above, , Q.52, , (B) 3, , tan–1(tan3x), ,  (1  x 2 ), (A), , dx, p 2  q 2 (tan 1 x ) 2, , +c, , cos (e, , x, , ), , dx equals-, , x, , x, , x, , )+c, , )+c, , ax 2  b, c 2 x 2  (ax 2  b) 2, , (B) sin (e, , x, , (D) – sin (e, , )+c, x, , )+c, , dx is, , , , cos x, 1  sin x, , dx is equal to-, , 1 sin x + c, , (C) 2 1 sin x + c, (D) 2 1 sin x + c, , =, Q.53, , sin 2 x, ,  a 2 sin 2 x  b 2 cos2 x dx is equal to(A), , p 2  q 2 (tan 1 x ) 2 ] + c, , (B), , 2 2, (C), (p + q2 tan–1 x)3/2 + c, 3q, , 1, b  a2, 2, , 1, a  b2, 2, , log (a2 sin2 x + b2 cos2 x ) + c, log (a2 sin2 x + b2 cos2 x ) + c, , (C) log (a2 sin2 x – b2 cos2 x) + c, , (D) None of the above, PHOTON, , x, , (B), , 1, log q tan 1 x  p 2  q 2 (tan 1 x ) 2  + c, , , q, , (B) log [q tan–1 x +, , x, , (A) 1 sin x + c, , 1, (C) tan–1(tan3x) + c (D) None of these, 3, , Q.47, , , , e, , (D) None of these, , dx is equal to+c, , (B) tanp+1x + c, ,  ax  b / x , (C) cos–1 , +k, c, , , , 2, , tan x sec x, , tan p 1 x, +c, p 1, ,  ax 2  b / x 2 , + k, (B) sin–1 , , , c, , , ,  tan(3x – 5) sec (3x –5) dx equals-, , Q.45, , 3, 1, sin n e x + c, 3, ,  ax  b / x , (A) sin–1 , +k, c, , , , (B) e–x + c, (D) 1, , (A) sin x + c, (C) ex + c, , (D) –, ,  cosp2 x dx equals-, , (A) 2 sin (e, , Q.51, ,  1  x  2!  3!  ....  dx, , Q.44, , 3, 1, sin e x + c, 3, , (C) sin (e, , (A) log (x + sin x) + c, (B) log(1+ cos x) + c, (C) log (1–sin x) + c, (D) None of these, , 3, , (B) 3 sin e x + c, , (C) (p +1) tanp+1 x + c (D) None of these, , 1  cos x,  x  sin x dx equals-, , Q.43, , 3, , cos (e x ) dx equals-, , sin p x, , (A), , 1, (A) (tan–1x3)2 + c, 3, , e, , 3, , (C), , (B) – cot + c, , (/3) + c, , 2 x3, , (A) sin e x + c, , Primitive of (sec/ tan2) is (A), , x, , (D) None of these, INDEFINITE INTEGRATION, , 23

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Q.54, , ( x  1) ( x  log x) 2, dx equalsx, (A) 3 (x + log x)3 + c (B) (x + log x)3+ c, , , ,  tan, , , , Q.61, 3, , 2x sec 2x dx equals-, , (A) sec3 2x – 3 sec 2x + c, (B) sec3 2x + 3 sec 2x + c, , (C) e, Q.62, , 1, (D) [sec3 2x + 3 sec 2x] + c, 6, 1, , Q.57, , 1, , (B) – cot (1/x) + c, (D) x cos (1/x) + c, , x2 , , +c, , 1, x, , , , x tan, , x, , 1 x4, , Q.63, , dx equals-, , Q.64, , x, , dx, x 4 1, , , , (A), , (C), , sec x 3 dx equals-, , equals1, sec–1 x2 + c, 2, (D) cosec–1 x 2 + c, , (B), , 4  x4, , dx is equal to-, , x2, +c, 2, , 1, x2, sin–1, +c, 2, 2, , (B) cos–1, (D), , x2, +c, 2, , 1, x2, cos–1, +c, 2, 2, , sin 2 x, ,  1  sin 2 x dx is equal to1, log (1+sin2 x) + c, 2, (D) tan–1 (sin x) + c, , (A) log (1+ sin 2 x) + c (B), , 1, log (sec x3 + tan x3) + c, 3, , (C) log sin 2 x + c, , (B) log (sec x3 + tan x3) + c, 1, (C) log (sec x3 – tan x3) + c, 3, , x, , (A) sin–1, , Q.65, 2, , +c, , 2, , 1, (tan–1 x2)2 + c, 2, (C) (tan–1 x2)2 + c, (D) None of these, , x, , x2, , 3, , (B), , Q.59, , (D) e, , +c, 1, ,  cos x dx is equal to-, , (A) sec–1 x2 + c, , 1, (tan–1 x2)2 + c, 4, , (A), , 1, x, , x2 , , +c, , (C) 2 sec–1 x2 + c, Q.58, , x, , 1, (D) cos x + cos3 x + c, 3, , (C) (1+ log x)3 + c, (D) None of these, 1, , (B) e, , 1, (C) sin x – sin3 x + c, 3, , , , 1, (1+ log x)3 + c, 3, , 1, x, , 1, (B) sin x + sin3 x + c, 3, , (1  log x ) 2, dx equalsx, (A) 3 (1 + log x)3 + c, (B), , x, , 1, (A) cos x – cos3 x + c, 3, ,  x 2 sin x dx equals(A) x sin (1/x) + c, (C) cos (1/x) + c, , 1, , 1   x x , ,  1  x 2  e  dx is equal to-, , (A) e, , 1, (C) [sec3 2x – 3 sec 2x] + c, 6, , Q.56, ,  cos x cot (sin x) dx equals(A) log cos (sin x) + c (B) log sin (sin x) + c, (C) – log cos (sin x) + c (D) – log sin (sin x) + c, , 1, (x + log x)3 + c (D) None of these, 3, , (C), Q.55, , Q.60, , Q.66, , 1  tan 2 x,  1  tan x dx is equal to-, , (A) –log (1 – tan x) + c, , (D) None of these, , (B) log (2 + tan x) – c, (C) log (1 – tan x) – c, (D) log (1 + tan x) + c, , PHOTON, , INDEFINITE INTEGRATION, , 24

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x3, , , , Q.67, , 1 x, , 8, , Q.72, , dx equals-, , (B), , 1, (C), sin–1 x2 + c, 4, , 1, (D), sin–1 x4 + c, 4, , xx x, , (C) 4 1 x + c, Q.73, , (A), , , , (0 < x < 1) then f ( x ) dx is equal to, , 1, 1, cot3 x – cot5 x + c, 3, 5, , (A) – 1 x 2, , 1, 1, (B) – cot3 x + cot5 x + c, 3, 5, , (C) –, , (C), , 1, 1, cot3 x – cot5 x + c, 3, 5, , Q.74, , (D) None of these, ,  1  2 tan 2 x dx is equal to(A), , 1, log (cos2 x + 2 sin2 x ) + c, 2, , (B), , 1, log (2cos2 x + sin2 x ) + c, 2, , (B), , 1, , (D), , x 2 1, , 1 x2, 1, 1 x2, , dx, ,  x log x . log (log x) is equals to -, , Q.75, , , , The value of (1+ tan x)3/2 sec2 x dx is-, , 1, (C) log (cos2 x + 2 sin2 x ) + c, 4, , (A), , 2, (1+ tan x)1/2+ c, 5, , (D) None of these, , (B), , 5, (1+ tan x)5/2 + c, 2, , , , 2  sin 3x . cos 3x dx =, , (C), , 2, (1+ tan x)5/2 + c, 5, , (A), , 2, (2 + sin 3x)1/2 + c, 9, , (D), , 2, (1+ tan x)1/2 + c, 3, , (B), , 2, (2 + sin 3x)2/3 + c, 3, , Q.70, , 1, , (A) log (x log x) +c, (B) log (log x) +c, (C) x log (log x) +c, (D) log({log(log x)} +c, , tan x, , Q.69, , (D) 2 x  x x + c, , n , , cos x, ,  sin 6 x dx equals-, , 1 x + c, , If f(x) = lim [2x + 4x3 + …..+ 2nx2n–1],, , 2, , Q.68, , equals-, , (A) log x  x x + c (B), , 1, sin–1 x3 + c, 4, , (A) sin–1x4 + c, , dx, , , , Q.76, , , , sec4 x, , dx is equal to-, , tan x, , (C), , 2, (2 + sin 3x)3/2 + c, 3, , (A), , 2, 5, , tan x (5 + tan2 x) + c, , (D), , 2, (2 + sin 3x)3/2 + c, 9, , (B), , 1, 5, , tan x (5 + tan2 x) + c, , (C), , 2, 5, , tan x (3 + tan2 x) + c, , , , Q.71, , sin x  cos x, , ,   3 , esin x cos x dx = if x  , , 1  sin 2x,  4 4 , , , (A) esin x + c, (C) esin x + cos x + c, , PHOTON, , (B) esin x – cos x + c, (D) ecos x – sin x + c, , INDEFINITE INTEGRATION, , (D) None of these, , 25

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Question, based on, , Integration by Parts, , x2, , , , Q.77, , 1 x, , 3, , Q.82, , 2, 3, , 1 x 3 + c, , (B) –, , 2, 3, , 1 x 3 + c, , (C), , 1, 3, , 1 x 3 + c, , (D) –, , 1, 3, , 1 x 3 + c, , , , (B) log (e/x2) + c, (C) log (x2/e) + c, (D) log x. log (e/x) + c, Q.83, , x 1, equalsx 1, , 1, x, , log (log x ), dx equalsx, ,  log x , (A) log x log ,  +c,  e , , dx equals-, , (A), , Q.78, , , , x e, , 3 x2, , (A), , 2, 1 2, (x + 1) e x + c, 2, , (B), , 2, 1 2, (x –1) e x + c, 2, , (A) log x  x 2  1 + sec–1x + c, (B) log x  x 2  1 – sec–1x + c, , dx is equal to-, , 2, 1, (1– x2) e x + c, 2, (D) None of these, , (C), (C) log x  x 2  1 – sech–1x + c, (D) None of these, Q.84, , , , Q.79, , x, , 1 x2, , dx =, , 1 x2, , (A), , 1, [sin–1x2 + 1 x 4 ] + c, 2, , (B), , 1, [sin–1x2 + 1 x 4 ] + c, 2, , (C) sin–1 x2 +, , x  sin x, ,  1  cos x dx =, (A) x cot, (C) cot, , Q.85, , 1 x 4 + c, , x, , , , Q.80, , x (1  x ), , (A) tan–1, (B), , is equal toQ.86, , dx, ,  x, , 1, tan4 x3 + c, 8, , 1, tan4 x3 + c, 12, , (D) None of these, ,  x log (1 + x)2 dx is equal to1, [2(x2 – 1) log (1+ x) – x2 + 2x] + c, 4, , 1, [2(x2 – 1) log (1+ x) – x2 – 2x] + c, 4, 1, (C) [2(x2 – 1) log (1+ x) – x2 – 2x] + c, 4, (D) None of these, , (B), , x+c, , is equal tox, , (A) log (1  x ) + c, , tan 3 x 3sec2 x 3 dx is equal to-, , (C), , x+c, , (C) 2 tan–1 x + c, , Q.81, , (D) None of these, , (B), , (A), , (D) 2 cot–1, , x, +c, 2, , x, +c, 2, , 1, tan4 x3 + c, 4, , x+c, , cot–1, , (B) – x cot, , (A), , (D) sin–1 x2 + 1 x 4 + c, dx, , 2, , x, +c, 2, , Q.87, , (B) log ( x  x ) + c, , The value of, , e, , x, , (cot x + log sin x)dx is-, , (A) ex log sin x + c, (C) ex log tan x + c, , (C) 2 log ( x  x ) + c, , (B) ex log cos x + c, (D) – ex log cos x + c, , (D) 2 log (1  x ) + c, PHOTON, , INDEFINITE INTEGRATION, , 26

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 sin–1(3x – 4x3) dx is equal to -, , Q.88, , Q.94, , (A) x sin–1x + 1 x 2 + c, (B) x sin–1x – 1 x 2 + c, (C) 2 [x, , sin–1x, , + 1 x ] + c, (C) –ex, Q.95, , ex, , (A) log sec x + c, (B) ex log tan x + c, (C) ex log (tan x + sec x) + c, (D) None of these, ,  (e, , logx, , Q.96, log x, ,  (1  logx) 2 dx equals-, , Q.91, , (A), , x, +c, 1  logx, , (C) –, , Q.92, , x, +c, 1  logx, , (B), , 1, , 1, +c, 1  logx, , (D) –, , 1, +c, 1  logx, , 1 xn, , +c, , 1 xn, , (D) –ex, , (A), , 1, [sec2x – tan x] + c, 2, , (B), , 1, [x sec2 x – tan x] + c, 2, , (C), , 1, [x sec2x + tan x] + c, 2, , (D), , 1, [sec2 x + tan x] + c, 2, , 1 xn, , +c, ,  cos (log x) dx is equal to-, , (B) –, , 2, x, 1, log (x2 + a2) + tan–1 + c, a, a, x, , cos (log x – /4) + c, , 2, , Q.97, , log x  1, ,  (log x ) 2, (A), , dx equals-, , x, +c, log x, , (C) –, , log (x2 + a2) dx =, 2, x, 1, log (x2 + a2 ) + tan–1 + c, a, a, x, , x, , (D), , Q.98, , (A), ,  sin, , (B), , x, +c, log x, , 1 , , 2x, , 1 x2, , x, (log x ) 2, , +c, , (D) None of these, , ,  dx equals, , (A) x tan–1 x + log (1+ x2) + c, (B) x tan–1 x – log (1+ x)2 + c, (C) 2x tan–1 x – log (1+ x2) + c, (D) None of these, Q.99, , 2, x, 1, (C) – log (x2 + a2) – tan–1 + c, a, a, x, (D) None of these, PHOTON, , 1 xn, , x, cos (log x – /4) + c, 2, x, (B) cos (log x +/4) + c, 2, x, (C), cos (log x + /4) + c, 2, , 1, ,  x2, , Q. 93, , 1 xn, , +c, , (B) ex, , (A), , 1, dx equalscot, x, 1, (A) x tan–1 x + log (1+ x2) + c, 2, 1, (B) x cot–1 1/x –, log (1+ x2) + c, 2, 1, (C) x cot–1 1/x + log (1+ x2) + c, 2, (D) None of these, , , , 1 xn, , ,  dx, , , ,  x sin x sec3 x dx equals-, ,  sinx) cos x dx equals-, , (A) x sin x + cos x + (1/2) cos 2x + c, (B) x sin x – cos x + (1/4) cos 2x + c, (C) x sin x + cos x – (1/4) cos 2x + c, (D) None of these, , +c, , 1 xn, , 2, ,  ex[log (sec x + tan x) + sec x] dx equals-, , Q.90, , 1 xn, , (A) ex, , (D) 3 [x sin–1x + 1 x 2 ] + c, Q.89, , , ,  1  nx n 1  x 2n, ex ,  (1  x n ) 1  x 2 n, , , INDEFINITE INTEGRATION, , e, , x, , [tan x – log cos x] dx, , = f(x) log sec x + c then range of f(x) is, (A) R, (B) R – {0}, (C) R+, (D) None of these, 27

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Question, based on, , Integration of rational function, , Q.104, ,  log x  2  5 , log ,  +c, 2 5,  log x  2  5 ,  log x  2  5 , log ,  +c, 5,  log x  2  5 , , 1, , (B), ,  log x  2  5 , log ,  +c, 2 5,  log x  2  5 , ,  2x 2  2x  3, , Q.105, , The value of, , (B), ,  2x  1 , 1, 5,  + c, log (2x2– 2x + 3) –, tan–1 , 4, 2,  5 , , (B), ,  2x  1 , 3, 5,  + c, log (2x2– 2x + 3)+, tan–1 , 4, 2,  5 , , 3, 5,  4x  2 , log(2x2– 2x + 3)+, tan–1 , +c, 4, 2,  5 , (D) None of these, (C), , x x2, , (A), , dx equals-, , (A), , (1  x ), 2, , +C, , dt, ,  t 2  2xt  1,  tx, tan–1 , , 2,  1 x, , 1, 1 x2, , (x2 > 1) is,  +c, , , ,  t  x  x 2 1 , +c, log , , , 2, 2, 2 x 1,  t  x  x 1 , , 1, , 1, log (t2 + 2xt + 1) + c, 2, (D) None of these, , Q.106, , , , 4x 2  x  1, x3 1, , dx equals-, , (A) log {(x3 –1)/(x–1)} + c, (B) log {(x – 1)/ (x3 – 1)} + c, , dx =, , x 1 x, – +c, x 1 2, , x 2 1, , (C), , 3, , (A) log, , (C) log {(x3 – 1) (x – 1)} + c,  x 1  x 2, (B) log , +c, +,  x 1 2, ,  x 1 x2,  x 1  x 2, (C) log , + c (D) log , +c, +, –,  x 1  2,  x 1 2, , (D) None of these, Q.107, , x, ,  x 4  x 2  1 dx equals(A), , x, ,  (x 2  1) (x 2  2) dx equals-, , Q.103, , tan–1, , 2 x  1 , +C, 2 x  1 , 2 x  1 , +C, 2 x  1 , , 2, 2x, (D) None of these, ,  log x  2  5 , log ,  +c, 5,  log x  2  5 , , 3x  1, , , , 1, , 1, , (D), , Q.102, , (C), , 1, , (C), , Q.101, , 1, , (A), , 1, , (A), ,  x 4  1 dx equals-, ,  x2 , log ,  x2 , 2 2, , , 1, x2 , (B), log  2, x , 2 2, , , dx, ,  x [(log x) 2  4 log x – 1] =, , Q.100, , x 2 1, , The value of, ,  x 2 1 , 1, + c, (A), log  2,  x 2, 2, , , , (B), ,  x2  2 , 1, + c, (B) log  2,  x 1 , 2, , , , (C), ,  x 2 1 , + c, (C) log  2,  x 2, , , , (D), ,  2x 2  1 , + c, tan–1 , , 3, 3 , , , 1, , 1, tan–1, 3, ,  2x 2  1 , , + c,  3 , , , ,  2x 2  1 , + c, tan–1 , , , 3, 3, , , , 2, ,  2x 2  1 ,  +c, tan–1 ,  3 , 3, , , , 1, ,  x2  2 , + c, (D) log  2,  x 1 , , , , PHOTON, , INDEFINITE INTEGRATION, , 28

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Q.108, , If, , dx, ,  x2  x3 =, , x, A, + B n, +C, x, 1, x, , 1, 2, (D) None of these, , 1, ,B=1, 2, (C) A = –1, B = –1, , (B) A = 1, B = –, , (A) A =, , dx, ,  (x 2  1) (x 2  4), , Q.109, , Q.113, , 1, 1, (C) tan–1 x + tan–1 x/2 + c, 3, 3, , Q.114, , (A), , x 2  3x, , dx is equal to-, , 1 2, x – 3x + 2 log x + c, 2, , dx, , Q.115, ,  x 1 , (B) log , + c,  2x  1 , ,  2x  1 , 1, 1,  x 1 ,  +c (D) log , log ,  +c, 3, 3,  2x  1 ,  2( x  1) , , dx, ,  x(x n  1), , Q.112, , cosx, ,  (1  sinx)(2  sinx) dx equals-, , 1,  1  sin x , log , + c, 2,  2  sin x , , dx, ,  x (x 4  1) equals(A), ,  x4 , 1,  +c, log  4,  x 1 , 4, , , , (B), ,  x 4 1 , 1, log  4  +c,  x , 4, , , ,  x 4 1 , (C) log  4  +c,  x , , , , dx is equal to-, ,  xn , 1, + c, (A) log  n,  x 1 , n, , , , (D) None of these, , Q.116, ,  x 1, 1, log  n  + c,  x , n, , , n, , (B), , 1, log (x4 + 1) + c, 4, , (D) None of these, , is equal to-, ,  2x  1 , (A) log , + c,  x 1 , , (C), , (C), , (C), , 1, (C) x2 – 3x – 2 log x + c, 2, (D) None of these, ,  2x 2  x  1, ,  x4 , 1, + c, log  4,  x 1 , 4, , , ,  1  sin x , (B) log , + c,  2  sin x , , 1, (B) x2 + 3x + 2 log x + c, 2, , Q.111, , (B), ,  2  sin x , (A) log , + c,  1  sin x , , (D) tan–1 x – 2 tan–1 x/2 + c, , , ,  x4 1, 1, log  4  + c,  x , 4, , , , (D) None of these, , 1, 1, (B) tan–1 x – tan–1 x/2 + c, 6, 3, , x 3  7x  6, , (A), , is equal to-, , 1, 1, (A) tan–1 x – tan–1 x/2 + c, 3, 3, , Q.110, , dx, ,  x (x 4  1) is equal to-, , , , ( x 3  8) ( x  1), , (A), , x 2  2x  4, , dx equals-, , x3 x 2, +, – 2x + c, 3, 2, ,  xn , +c, (C) log  n,  x 1 , , , , (B) x3 + x2 – 2x + c, , (D) None of these, , (C), , 1 3, (x + x2 – x) + c, 3, , (D) None of these, PHOTON, , INDEFINITE INTEGRATION, , 29

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Question, based on, , Integration of irrational function, , dx, , , , Q.117, , 5x  6  x 2, , Q.122, , , , dx, 2x  x  2, , equals-, , 4x  1, , (A) log, , (B) cos–1 (2x + 5) + c, (C) sin–1 (2x – 5) + c, , 2x  3, x 1, , (C), , (D), , x 2  1 + 3 log x  x 2  1 + c, , 1 x2, 1 x, , 2, , 4x  1, 2x 2  x  2, +c, , 4, 2, , 1, 3 –1, sin x – x 1 x 2 + c, 2, 2, , (B), , 1, 3 –1, sin x + x, 2, 2, , , , Question, based on, , 1 x 2 + c, , Q.124, , 1, [sin–1 x – x 1 x 2 ] + c, 2, (D) None of these, , , , x 2  x 1, , , , x  x 1 + c, , (A), (C), , 1, 2, , (B) 2 x  x  1 + c, 2, , x 2  x  1 + c (D) None of these, , dx, x (1  x ), , 4x  1, , , , 15, , dx, 2  3x  x 2, , 4x 2  8x  14, +c, 15, , is equal to-, , equals-, , Q.125, , (A) sin–1(1– 2x) + c, , 1  sin x, ,  2x  3 , ,  + c,  17 , , =, ,  x , 2 log tan    + c, 4 8, , (B), ,  x , 2 log tan    + c, 4 8, , (C), ,  x , 2 log sin    – c, 4 8, , (D), ,  x , 2 log sec    – c, 4 8, , The value of, (A), , (B) log 1  2x  4x 2  4x  2 + c, , dx, , (A), , dx equals-, , 2, , log, , Integration of trigonometric function, , (C), , 2x  1, , 1, ,  2x  3 ,  2x  3 ,  + c (D) cos–1 ,  + c, (C) sin–1 ,  17 ,  17 , , dx equals-, , (A), , sin x, ,  sin x  cos x dx equals-, , 1, 1, x + log (sin x – cos x) + c, 2, 2, , 1, 1, x – log (sin x – cos x) + c, 2, 2, (C) x + log (sin x + cos x) + c, (D) None of these, , (B), , (C) sin–1(2x – 1) + c, (D) log 2x  1  4x 2  4x + c, , PHOTON, , 4x 2  8x  14, +c, 15, ,  2x  3 ,  + c (B) sec–1, (A) tan–1 ,  17 , , (D) None of these, , , , log, , 2, , (C) 2 x 2  1 + 3 log x  x 2  1 + c, , Q.121, , 1, 2, , dx is equal to-, , Q.123, , Q.120, , , , 2, , (B), , , , 4x 2  8x  16, +c, 15, , 15, , (A) 2 x 2  1 + 3 log x  x 2  1 + c, , Q.119, , 4x  1, , (B) log, , (D) log 2X – 5  4X 2 – 20 X  24 + c, , , , , , 15, , (A) sin–1 (2x + 5) + c, , Q.118, , equals-, , 2, , INDEFINITE INTEGRATION, , 30

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dx, ,  a sin x  b cos x equals-, , Q.126, , Q.131, ,  1, b , log  tan  x  tan 1  + c, 2, a , , a 2  b2, , (A) x + 2 [1 + tan (x/2)]–1 + c, ,  , b , log tan  x  tan 1  + c, a ,  , a b, , (C) x – 2 [1 + tan (x/2)]–1 + c, , 1, , (A), , (B) x + [1 + tan (x/2)]–1 + c, , 1, , (B), , 2, , 2, ,  1, b , log  tan  x  tan 1  + c, 2, 2, 2, a , , , a b, (D) None of these, , (D) None of these, , 1, , (C), , Q.132, , dx, ,  1  2 sin x  cos x equals-, , Q.127, , sin x dx, ,  1  sin x equals-, , (A) log (1 + 2 tan x/2) + c, (B) log (1 – 2 tan x/2) + c, 1, log (1 + 2 tan x/2) + c, 2, 1, (D) log (1 – 2 tan x/2) + c, 2, cos 2 x,  (sin x  cos x ) 2 dx is equal to-, ,  sin3 x dx is equal to(A), , 1, cos3 x + cos x + c, 3, , (B), , 1, cos3 x – cos x + c, 3, , (C), , 1, (cos3 x + cos x) + c, 3, , (C), , Q.128, , Q.129, , (A) log (sin x – cos x) + c, (B) log (cos x – sin x) + c, (C) log (sin x + cos x) + c, (D) none of these, dx,  9 sin 2 x  4 cos2 x is equal to-, , (D) None of these, Q.133, , , , 2, , (B), , (C) 6, (D), , (C), , Q.134, ,  x , log tan    + c, 2 8, ,  x , log tan    + c, 2 4, 2, , 1, , 3, ,  tan x  + c, 2, , , , 2, , 3, (A) tan–1(3 tan x/2) + c, 2, 2, (B) tan–1(3 tan x/2) + c, 3, (C) tan–1(3 tan x/2) + c, , (D) None of these, , INDEFINITE INTEGRATION, , 3, , (A), , 1 3, 1, sin x – sin5 x + c, 3, 5, , (B), , 1, 1, cos3 x – sin5 x + c, 3, 5, , (C), , 1, 1 3, sin x – sin5 x + c, 5, 3, , (D), , 1, 1, tan3 x – sin5 x + c, 3, 5, , dx, , PHOTON, ,  x , log tan    + c, 2 4, ,  sin x cos x dx is equal to-, ,  5  4 cos x equals-, , Q.130, , equals-, , (D) None of these, , , tan–1, , 1, tan–1, 6, , 1, 2, , 2, ,  tan x  + c, 3, , 3, ,  tan x  +c, 2, , , 1, , (A), , 3, , (A) tan–1  tan x  + c, 2, , , (B) tan–1, , dx, 1  sin 2x, , 31

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x2, , , , Q.135, , 1 x, , dx is equal to-, , Q.138, , (C) log (1+ ex) – x + e–x + c, (D) log (1+ ex) + x + e–x + c, , 1, (B), log (1 – x) (3x2 + 4x + 8), 15, , 2, 15, , Q.139, , 1 x (3x2 + 4x + 8), , dx, ,  (1  e x )(1  e x ) equals ex 1 ,  + c (B) log, (A) log  x,  e 1, , , , (D) None of these, Some integration of different, Expression of ex, , Question, based on, ,  1  e x dx =, (A) log (1+ ex) – x – e–x + c, (B) log (1+ ex) + x – e–x + c, , 2, (A) (1 – x)3/2 (3x2 + 4x + 5), 3, , (C), , ex, , (C), , ex 1, ,  e x  1 dx is equal to-, , Q.136, , (A) log (ex + 1) + c, , Q.140, , (B) log (ex – 1) + c, , (D) None of these, , a, , x, , , , (A) 2  e x  1  log e 2  e  x  1  + c, , , , , , Q.141, , x, , , , (B) 2  e x  1  log e 2  e  x  1  + c, , , , , , (C) 2  e x  1  sin 1 (e  x / 2 ) + c, , , (D) None of these, , PHOTON, , x, a, a,  x , log  e  + k (B) log  b  ce  + k, x, x, , ,  e, , b, b,  b  ce , , , , (C) c log (b + cex) + k (D) None of these, , e x  1 dx is equal to -, , Q.137, ,  ex 1 ,  x, , 1,  + c (D) 1 log  e  1  + c, log  x,  e 1 ,  ex 1 , 2, 2, , , , , ,  b  ce x dx is equal to(A), , (C) 2 log (ex/2 + e–x/2) + c, ,  ex 1 , , +c,  ex 1 , , , , INDEFINITE INTEGRATION, , , , dx, 1  e 2x, , =, , (A) log  e  x  e  2 x  1 , , , (B) log  e  x  e  2 x  1 , , , (C) log  e x  e 2 x  1 , , , (D) log  e x  e 2 x  1 , , , , 32

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LEVEL- 2, x5, ,  1  x12 dx is equal to-, , Q.1, , (A) tan–1x6 + c, (C), , , , Q.2, , Q.8, , (B) 2 tan–1x6 + c, , 1, tan–1x6 + c, 6, , x, , , , a  x3, 3, , dx is equal to-, , x, (A) sin–1  , a, , (D) None of these, , 3/ 2, , 3, x, (C) sin–1  , 2, a, , sin x dx is equal to-, , (A) 2 (sin x – cos x ) + c, , +c, 3/ 2, , (B), , 2, x, sin–1  , 3, a, , 3, x, +c (D) sin–1  , 2, a, , dx, , (B) 2 (sin x + cos x ) + c, ,  sin (x  a) cos (x  b), , (C) 2 (sin x – x cos x ) + c, , (A) cos (a – b) log, , sin ( x  a ), +c, cos ( x  b), , (B) sec (a – b) log, , sin ( x  a ), +c, cos ( x  b), , (C) sin (a – b) log, , cos ( x  a ), +c, sin ( x  b), , (D) 2 (sin x +, , Q.9, , x cos x ) + c, , dx, ,  x  x log x is equal to-, , Q.3, , (A) log x + log (log x) + c, (B) log log (1+ log x) + c, , (D) cosec (a – b) log, , (C) log (1+ log x) + c, (D) None of these, , , , Q.4, ,  x , e x / 2 sin    dx is equal to2 4, , Q.10, , is equal to-, , cos ( x  a ), +c, sin ( x  b), ,  x cos x dx is equal to2, , (A), , 1, x2 1, – x sin 2x – cos 2x + c, 8, 4 4, , (B), , 1, x2 1, – x sin 2x + cos 2x + c, 8, 4, 4, ,  {sin (log x)  cos (log x)} dx is equal to-, , (C), , (A) sin (logx) + c, (C) x sin (log x) + c, , 1, 1, x2, + x sin 2x – cos 2x + c, 8, 4, 4, , (D), , 1, 1, x2, + x sin 2x + cos 2x + c, 8, 4, 4, , x, , (tan–1 x + cot–1 x) dx is equal to-, , (A), , x 52, (tan–1 x + cot–1 x) + c, 52, , (D) x(log10 x – log10 e) + c, , (B), , x 52, (tan–1 x – cot–1 x) + c, 52, ,  [(log 2x)/x] dx equals-, , (C), , x 52 , + +c, 52, 2, , (A) ex/2 sin x/2 + c, (C), Q.5, ,  log, , Q.6, , (B) ex/2 cos x/2 + c, , 2 ex/2 sin x/2 + c (D), , 10, , 2 ex/2 cos x/2 + c, , (B) cos (log x) + c, (D) x cos (log x) + c, , x dx is equal to-, , (A) log10 x + c, , Q.11, , (B) x log10 x + c, , 51, , (C) x (log10 x + log10 e) + c, , Q.7, , (A) x log 2x + c, (B) (log x log 2x)/2 + c, (C) (log x log 4 x) /2 + c, , (D) –, , x 52 , + +c, 104, 2, , (D) None of these, PHOTON, , INDEFINITE INTEGRATION, , 3/ 2, , 33, , +c, 2/3, , +c

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sin 8 x  cos8 x, , (A) sin 2x + c, 1, (C), sin 2x + c, 2, , Q.13, , Q.18, ,  1  2 sin 2 x cos2 x dx is equal to-, , Q.12, , If x = f (t) cos t +f (t) sin t,, , Q.19, 1/ 2, , (A) f (t) + f (t) + c, , (B) f  (t) + f  (t) + c, , (C) f (t) + f  (t) + c, , (D) f (t) – f  (t) + c, , , , If I = ex sin 2x dx, then for what value of k,, , 1, (B) – cos 4x + c, 2, , 1, cos 4x + c, 8, , (D) None of these, , (C) –, , Q.20, , (C), , x, ( x  1), , 2, , ex + c, , x 1 x, e +c, x 1, , (D), , x x, e +c, x 1, , –1, , x, , (D) None of these, , dx is equal to-, , (B), , 3, sin–1 (cos3/2 x) + c, 2, , (C), , 2, cos–1 (cos3/2 x) + c, 3, , If, , , ,  1  x 3  1,  + b, then= a log ,  1  x 3  1, x 1 x3, , , , dx, , 1, 3, , (C) a = –, , Q.22, Q.17, , (B) x 2e tan, , +c, , 2, sin–1 (cos3/2 x) + c, 3, , (A) a =, , (B), , x, , ,  dx is equal to, , , (A), , x 1, ,  (x  1) 2 ex dx equal tox 1 x, e +c, x 1, , –1, , 1  cos3 x, , 2, , (A), ,  1 x, , cos x  cos3 x, , , , 2, , (D) None of these, , Q.21, Q.16, , x,  1 x  x, 2, , , 1, sin4 x + c, 4, , (D) None of these, , 1 tan –1 x, +c, e, x, , (C), , cos 4 x  1,  cot x  tan x dx equals-, , 1, (A) – cos 4x + c, 2, , –1, , (A) xetan, , kI = ex (sin 2x – 2 cos 2x) + constant(A) 1, (B) 3, (C) 5, (D) 7, Q.15, , e tan, , dt, , is equal to-, , Q.14, , , , (B) –, , 1, cos4 x + c, 4, , (C) –, , (D) – sin 2x + c, ,  dx  2  dy  2 ,   dt    dt  , , , , 1 sin x, e +c, 4, , (A), , 1, (B) – sin 2x + c, 2, , y = –f (t)sin t+f (t)cos t, then, ,  cos3 x e log (sin x) dx is equal to-, , x, x, 3 tan  tan 3, 3, 3 dx is equal tox, 1  3 tan 2, 2, (A) –log |sec x| + c, (B) – log |(cos x)| + c, (C) sec2 x + c, , , , 1, 3, , 2, 3, , (D) a = –, , 2, 3, , x tan 1 x, ,  (1  x 2 )3 / 2 dx equals to(A), , (B), , (C), , (D) log |tan x| + c, , (B) a =, , x tan 1 x, 1 x2, , +c, , x  tan 1 x, 1 x2, tan 1 x  x, 1 x2, , +c, , +c, , (D) None of these, , PHOTON, , INDEFINITE INTEGRATION, , 34

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3 cos x  2 sin x, ,  4 sin x  5 cos x dx is equal to-, , Q.23, , Q.28, , 23, 2, x+, log (4 sin x + 5 cos x) + c, 41, 41, 23, 2, (B), x–, log (4 sin x + 5 cos x) + c, 41, 41, 23, 2, (C), x–, log (4 sin x – 5 cos x) + c, 41, 41, (D) None of these, , (A) –, , (A), , x2 1, ,  x 2 1 ,  +c, (A) log ,  x , , , ,  x 2 1 , + c, (B) – log ,  x , , , ,  x , (C) log  2, + c,  x 1 , ,  x , (D) – log  2, + c,  x 1 , , Q.29, , (C), , sin 3 (ax  b), +c, 3a, , , , x n 1, n 1, , (B), , 2 , x n 1 , log x , , n  1, n 1 , , (C), , x n 1, n 1, , 1  sec x dx equals-, , (B) –2 sin–1( 2 sin x/2) + c, , 1 , , log x , + c, n,  1, , , Q.30, , 1 , , 2 log x , + c, n,  1, , , 1 , x n 1 , log x , +c, n  1, n 1 , , (C), , x, ( + x) + c, 4, , sin x, , (A), ,  3  tan x ,  +c, log ,  3  tan x , 2 3, , , , (B), ,  3  tan x , +c, log ,  3  tan x , 2 3, , , , Q.31, , 4x  7, , (A) 2 log, , (x2, , 1, , 1, ,  3  tan x , +c, log ,  3  tan x , 3, , , , 1, , (D) None of these, , (D) None of these, ,  x 2  x  2 dx equals-, , Q.27, , x, x,  2 sin 2  1, 2, 2, ,  sin 3x dx is equal to-, , (C), , sec x, (B), +c, sec x  tan x, , x, (A) + c, 2, , 2 sin, , (D) None of these, ,  tan–1 (sec x + tan x) dx equals-, , Q.26, , sin 3 (ax  b), +c, 3a, , (C) 2 log, , (A), , (D), , cos3 (ax  b), +c, 3a, , (A) 2 sin–1( 2 sin x/2) + c, ,  xn log x dx equals-, , Q.25, , cos3 (ax  b), +c, 3a, , (B), , (D) –, ,  x(x 2  1) dx is equal to-, , Q.24, ,  cos2 (ax + b) sin (ax + b) dx equals-, ,  x 1 , + x – 2) – 3 log , + c, x2, ,  x 1 , (B) 2 log (x2 + x – 2) + 3 log , + c, x2, , 1  cos x, ,  cos x (1  cos x) dx is equal to(A) log (sec x + tan x) – 2 tan x/2 + c, (B) log (sec x + tan x) + 2 tan x/2 + c, (C) log (sec x – tan x) – tan x/2 + c, (D) None of these, ,  x 1 , (C) 3 log (x2 + x – 2) + 2 log , + c, x2, (D) None of these, PHOTON, , INDEFINITE INTEGRATION, , 35

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 ex, , Q.32, , 1 e x dx is equal to-, , Q.37, , x 5 dx, , , , 1 x3, , (A), , 2, (1 + ex)3/2 + c, 3, , (A), , 2, 3, , (1  x 3 ) (x2 + 2) + c, , (B), , 3, (1+ ex)3/2 + c, 2, , (B), , 2, 9, , (1  x 3 ) (x3 – 4) + c, , (C), , 2, 9, , (1  x 3 ) (x3 + 4) + c, , (D), , 2, 9, , (1  x 3 ) (x3 – 2) + c, , (, , tan x  cot x ) dx equals-, , (C) ex (1+ ex)3/2 + c, 2, (D) ex(1+ex)3/2+ c, 3, , e, , log[1 (1 / x 2 )], ,  [x 2  (1 / x 2 )] dx equals-, , Q.33, , (A) e log[1 (1/ x, , 2, , Q.38, )], , +c, , 1, , (C), , tan–1, , 2, , If, , du, dv, –v, +c, dx, dx, , Q.39, , (B) 2, , du dv, +, +c, dx dx, , (D) u, , du, dv, +v, +c, dx, dx, , (B), , Q.40, , cot x ) + c, , x, , 55, , x, , (log 5) 3, , +c, , (B), , 5x, , 55, , 5x, , (log 5) 3, , +c, , (D) None of these, ,  [1+ 2 tan x (tanx + sec x)]1/2 dx equals(A) log sec x + log (sec x + tan x) + c, (B) log sec x – (sec x + tan x) + c, (C) log (sec x + tan x) / sec x + c, (D) None of these, , {f(x)]3, , (D) {f(x)]2, , , , 5x, , (C) ( 55 log 5)3 + c, , Q.41, , x3 1, , , , dx, ( x  ) (  x ), , (A) 2 sin–1, , 1, log (x2 + 1) – tan–1 x + c, 2, 1, (B) x + log x + log (x2 + 1) – tan–1 x + c, 2, 1, (C) x – log x – log(x2 + 1) – tan–1 x + c, 2, (D) None of these, , (A) x – log x +, , PHOTON, , cot x ) + c, , 55 . 55 . 5x dx is equal to-, , (A), ,  x 3  x dx equal to-, , Q.36, , ( tan x +, , 1, , ( tan x –, 2, (D) None of these, ,  f(x) dx = f (x), then  {f(x)}2 dx is equal to, , 1, (A) {f(x)}2, 2, 1, (C) {f(x)]3, 3, , cot x ) + c, , 1, , (C) tan–1, ,  u dx 2 dx –  v dx 2 dx is equal to -, , (C) u, , ( tan x –, , 2, , d 2u, , (A) uv + c, , 2 tan–1, , (B), ,  x 2 1 , + c, (D) tan–1 ,  x 2 , , , , Q.34, , 1, 2, ,  x2 1, , + c,  x 2 , , , , d2v, , 2 tan–1, , (A), ,  x 2 1 ,  +c, (B), tan–1 ,  x 2 , 2, , , 1, , Q.35, , equals-, , INDEFINITE INTEGRATION, , (B), , , ( > ) equals-, , x, +c, , , 1, x, sin–1, +c, 2, , , (C) 2 sin–1, , x , +c, , , (D) None of these, 36

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, ,  sin tan, , Q.42, , 1, , , , (A), , ,  dx equals1  x 2 , x, , 1 x 2 + c, , (C) cos, , (B), , 2, , 1 x + c, , Q.46, , (A), (B), , x2  4 + C, , 1, x 4, , +C, , 2, , 2, , 1 x + c, , (C) 2 x 2  4 + C, , sin 2x, ,  sin 4 x  cos4 x dx is equal to-, , Q.43, , x 4, , w.r.t. (x2 + 3) is equal to-, , 2, , 1 2, x +c, 2, , (D) – cos, , 1, , Integral of, , (D) None of these, , 1, tan–1(tan2x) + c (B) 2cot–1(tan2x) + c, 2, (C) tan–1(tan2x) + c, (D) None of these, (A), , , , Q.44, , Q.47, , If f(x) is the primitive of, , sin x1/ 3 log (1  3x ), (tan 1 x ) 2 (e x, , 1/ 3, , x  0 then lim f (x) is:, x 0, , log (x + x 2  a 2 ) dx is equal to-, , (A) x log  x  x 2  a 2  + x 2  a 2 + C, , , , (A) 0, , (B) 3/5, , (C) 5/3, , (D) None of these, , (B) x log  x  x 2  a 2  – 2 x 2  a 2 + C, , , (C) x log  x  x 2  a 2  –, , , , x2  a2 + C, , (D) None of these, Q.45, , Integral of, , (A), , 1, 1  (log x ) 2, , w.r.t. log x is-, , tan 1 (log x ), +C, x, , (B) tan–1 (log x) + C, , tan 1 x, +C, x, (D) none of these, (C), , LEVEL- 3, PHOTON, , INDEFINITE INTEGRATION, , 37, ,  1), , ,

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Q.1, , 2x, , If , , 1 4x, , Q.6, dx = K, , sin–1, , (2x), , + C, then K is, , Q.2, , (B), , 1, log 2, 2, , 1, 2, , (D), , 1, log 2, ,  g(x) dx = g(x), then  g(x) {f(x) + f  (x)} is, , (C) A = 1, (D) None of these, Q.7, , If, ,  f (ax + b) {f(ax + b)}n dx is equal to, (A), , 1, {f(ax + b)}n+1+ C,  n except n = –1, n 1, , (B), , 1, {f(ax+b)}n+1+ C,  n, (n  1), 1, {f(ax+b)}n+1 + C,  n except n = –1, a ( n  1), , Q.8, , , , 1, , (A), , Q.9, , tan x, , (C), , 2, tan x, , Q.5, , +C, , (B) 2 tan x + C, , Q.10, , (D) –2 tan x + C, , The value of the integral , , log ( x  1)  log x, dx isx ( x  1), , Q.11, , 1, 1, [log(x+1)]2– (logx)2+log(x+1)log x +C, 2, 2, (B) –[log(x+1)]2 – (log x)2 + log(x+1)log x + C, , (A)–, , (B) B = –1/6, , (C) A = –1/3, , (D) (A) and (B), , If, , cos4 x, ,  sin 2 x dx = A cot x + b sin 2 x + C, , (A) A = – 2, , (B) B = – 1/4, , (C) C = – 3, , (D) (B) and (C), , 2x 2  3, ,  (x 2 1) (x 2  4) dx, , (A) (–1/2, 1/2), , (B) (1/2, 1/2), , (C) (–1, 1), , (D) (1, – 1), , , , d(cos ), , is equal to-, , 1  cos2 , , (A) cos–1  + C, , (B)  + C, , (C) sin–1  + C, , (D) sin–1(cos ) + C, , x, ,   log (1  cos x )  x tan 2  dx is equal to , , , (A) x tan, , x, 2, , (B) log (1 + cos x), , 1, (C) – [log (1+1/x)]2 + C, 2, (D) (A) and (C) is correct, , (C) x log (1 + cos x), (D) None of these, 2, , Q.12, , PHOTON, , x, +C,, 2, , x,  x 1, = a log ,  + b tan–1 , then (a, b) isx, , 1, 2, , , , dx is equal to-, , +C, , B tan–1, , (A) A = 1/3, , sin 3 x cos x, 2, , –1 x +, , x/2 + D, then -, , 1, (D), {f(ax + b)}n+1 + C,  n, a ( n  1), , Q.4, , 1, ,  (x 2  1) (x 2  4) dx =A tan, , then -, , (D) g(x) f2(x) + C, , (C), , 1 2, x, 2, , (B) g(x) = log x, , equal to(A) g(x) f(x) – g(x) f '(x) + C, (B) g(x) f '(x) + C, (C) g(x)f(x) + C, , Q.3, , (1 + 1/x) dx = f(x).log (x +1) + g(x)., , (A) f(x) =, , (A) log 2, , If, ,  x log, , x2 + Ax + C, then -, , equal to-, , (C), , If, , INDEFINITE INTEGRATION, , x  1 x ,  e  1  x 2  dx is equal to-, , 38

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 x  cos x , (B) n ,  +c, x, , , ,  1 x , (A) e x , +C, 1 x2 , ,  cos x , (A) n ,  +c,  x , ,  x 1 , (B) e x , +C, 1 x2 , , x, , , (C) n ,  + c (D) None,  x  cos x , , 1, , (C) ex., , +C, 1 x2, (D) None of these, ,  (x, , Q.13, , (A), , Q.17, , x2, 2, ,  a )( x  b ), 2, , 2, , 2, , dx, 2e x 1, , (A) sec–1, , dx is equal to-, , x, , 1 x,  a tan 1   C,  b tan, b, a, (b  a ) , 2, , Q.18, , , , sin 3 x, 3, , cos 2 x, , (A) 3, , x, , 1 x,  a tan 1   C,   b tan, b, a, b a , , 3, , 1, , 2, , 2, , 1– x7, ,  x (1 x, , 7, , ), , (B) nx, , Q.19, , 2, n (1 – x7) + c, 7, , (C) nx –, , cos3 x  1 + c, , Q.20, , sin 2x  2 | + c, , (B) log |sinx + cosx +, , sin 2x | + c, , , , cos x  sin x, , dx is equal to-, , sin 2x, , (C) log | sin x + cos x +, , 2  sin 2A | + c, , sin 2A | + c, , (D) cos–1(sin x + cos x) + c, Q.21, , (C) 2 (1+ 1 x 2 ) + c, (D) None, , PHOTON, , (A) log |sinx + cosx +, , (B) sin–1(sin x – cos x) + c, , 1, n (1+ 1 x 2 ) + c, 2, , x 2  cos 2 x, , dx equals-, , sin 2x, , (A) log |sinx – cos x +, , 1  x 2  (1  x 2 ) 3, , cos x  x sin x, , cos x  sin x, , (D) None of these, , x dx, , (B) 2 1  1  x 2 + c, , , , , , (C) log (sinx + cosx) – sin 2x  2 + c, , 2, n (1 + x7) + c, 7, , 2, (D) nx +, n (1 – x7) + c, 7, , Q.16, , dx equals-, , (D) None of these, , dx, , 2, (A) nx + n (1 + x7) + c, 7, , (A), , 2e x + c, , (B) 3 3 cos 2 x (7 cos2 x – 1) + c, (C) log, , , , (B) sec–1 ( 2e x ) + c, , 1, , cos x  cos2 x 1 + c, 7, , , (D) None of these, , Q.15, , 2e x + c, , 1, , 2, , x, , 1 x,  a tan 1   C, (B) 2,  b tan, 2, b, a, b a , , Q.14, , equals-, , (C) 2sec–1 ( 2 ex)+ c (D) 2 sec–1, , 1, , (C), , , , 4e x  6e  x, ,  9e, , dx =Ax + B log (9e2x–4) + c,,  4e  x, then values of A and B are(A) – 19/36, – 35/36 (B) 3/2, 35/36, (C) –3/2,–35/36, (D) –3/2, 35/36, If, , x, , dx, ,  Statement type Questions, , INDEFINITE INTEGRATION, , 39

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Each of the questions given below consists of, Statement-I and Statement-II. Use the following key, to choose the appropriate answer., (A), If both Statement-I Statement-II are true,, and Statement-II is the correct explanation, of Statement-I., (B), If both Statement-I and Statement-II are, true but Statement-II is not the correct, explanation of Statement-I, (C), If Statement-I is true but Statement-II is false, (D), If Statement-I is false but Statement-II is true., Q.22, , Statement- I:, , , , 5 – 2x, 2  2x – x 2, , dx, ,  x –1 ,  + c, = 2 2  2x – x 2 + 3 sin–1 ,  3 , , Statement- II:, , =, , x, 2, , , , a2 – x2 +, , 1, 2, , a – x2, , dx, , x, a2, sin–1, a, 2, , Observe the following statements:, Statement- I:, , , , Statement- I:, , x2 –1, x2, ,  x 2 1 , , ,  x ,  dx, e, ,  f (x)e, , Statement- II:, , Q.23, , Q.27, , , , f (x), , x9/ 2, 1 x, , 11, , =, , x 2 1, e x, , +c, , dx = ef(x) + c., , dx =, , 2, log, 11, , 11, , | x 2  1  x11 | + c, Statement- II:, Q.24, , dx, 1– x, , 2, , = log |x + 1 x 2 | + c, , Statement- I:, , , , 10 x 9  10 x log e 10, 10 x  x10, , Statement- II:, , Q.25, , , , Statement- I:, =, , dx = log |10x + x10| + c, , f ' (x), ,  f (x), , dx = log |f (x)| + c, ,  tan 3x tan 2x tan x dx, , n | sec 3x | n | sec 2x |, –, – n |sec x| + c, 3, 2, , Statement- II: tan 3x – tan 2x – tan x, = tan 3x tan 2x tan x, Q.26, , Statement- I:, , e, , x, , (sin x  cos x) dx  e x sin x  c, , Statement- II:, , e, , x, , (f (x)  f ' (x) dx  e x f (x)  c, , LEVEL- 4, PHOTON, , INDEFINITE INTEGRATION, , 40

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(Question asked in previous AIEEE and IIT-JEE), sin x dx, SECTION –A, , , , Q.1, , Q.6, , cos 2 x  1, dx =, cos 2 x  1, , (A) tan x – x + c, (C) x – tan x + c, Q.2, , If, , , , (C) (–sin , cos ), , (D) (–cos , sin ), ,  cos x  sin x is equal todx, , (C), , (D), , , , Q.4, , [AIEEE 2004], , (C), , x, log cot   + C, 2, 2, , , )|+c, 4, , (C) x – log | cos (x –, , , )|+c, 4, , (D) x + log | cos (x –, , , )|+c, 4, , Q.1, ,  x 3 , log tan    + C, 2, 2 8 , ,  (x  p), , dx, ( x  p) ( x  q), , is equal to[IIT -1996], , 1, , (A), ,  x 3 , log tan    + C, 2, 2 8 , , 1, , log x, (log x ) 2  1, , xe x, 1 x2, , +C, , +C, , (B), , (D), , equals-, , (C), x, x2 1, , +C, , x, (log x ) 2  1, , x q, +c, xp, , 2, pq, , [AIEEE-2005], , 1, ( x  p)( x  q), , +c, , (D) None of these, +C, , [AIEEE 2007], , 3 sin x, , 1, x  , log tan    C, 2,  2 12 , , 1, x  , (B) log tan    C, 2,  2 12 , , x  , (C) log tan    C,  2 12 , x  , (D) log tan    C,  2 12 , , xp, +c, x q, , 2, pq, , (B) –, , 2, ,  cos x , , PHOTON, , (B) x + log | sin (x –, , SECTION-B, , 1, , dx, , (A), , , )|+c, 4, ,  x , log tan    + C, 2, 2 8, , Q.2, Q.5, , 4, , (A) x – log | sin (x –, , 1, ,  (log x  1) , ,  dx is equal to1  (log x ) 2 , , (A), , is -, , [AIEEE 2008], , [AIEEE 2004], (B) (cos , sin ), , (B), ,  sin x   , , , sin x, dx = Ax + B log sin (x – ) + C,, sin(x  ), , (A) (sin , cos ), , (A), , 2, , (B) x + tan x + c, (D) – x – cot x + c, , then value of (A,B) is-, , Q.3, , The value of, , [AIEEE 2002], , ( x  1), ,  x (1  x e, , x 2, , ), , dx is equal-, ,  x ex, (A) log ,  1 x ex, , , , 1, +, +c,  1 x ex, , ,  x, (B) log ,  1 x ex, , , , 1, +, +c,  1 x ex, , ,  1 x ex, (C) log ,  x ex, , , , 1, +, +c,  1 x ex, , , [IIT- 1996], , (D) None of these, Q.3, INDEFINITE INTEGRATION, , cos x  sin x, ,  cos x  sin x, , (2 + 2 sin 2x) dx is equal to, , 41

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[IIT 1997], Q.7, , (A) sin 2x + c, , (B) cos 2x + c, , (C) tan 2x + c, , (D) None of these, , Let I =, J=, ,  ( 2 x  7), , Q.4, , dx, x 2  7 x  12, , is equal to[IIT 1997], , ex, ,  e 4x  e 2x  1 dx,, ex, ,  e 4x  e 2x  1 dx then for an arbitrary, , constant c, then the value of J – I equals, [IIT- 2008], , (A) 2 sec (2x – 7) + c, (B) sec–1 (2x – 7) + c, , (A), ,  e 4x  e 2x  1 , 1, + c, log  4 x,  e  e 2x  1 , 2, , , , 1, sec–1 (2x – 7) + 2, 2, (D) None of these, , (B), ,  e 2x  e x  1 , 1, + c, log  2 x,  e  e 2x  1 , 2, , , , (C), ,  e 2x  e x  1 , 1, +c, log  2 x,  e  ex 1 , 2, , , , (D), ,  e 4x  e 2x  1 , 1, + c, log  4 x,  e  e 2x  1 , 2, , , , –1, , (C), , , , x, ,  cos x log  tan 2  dx is equal to-, , Q.5, , [IIT-1998], x, , (A) sin x log  tan  + c, 2, , , (B) sin x log tan, , x, –x+c, 2, , x, , (C) sin x log  tan  + x + c, 2, , , (D) None of these, Q.6, , Let F(x) be an indefinite integral of sin 2x., [IIT- 2007], STATEMENT-1: The function F(x) satisfies, F(x + ) = F(x) for all real x., because, STATEMENT-2: sin2(x + ) = sin2x for all real x., (A) Statement-1 is True, Statement-2 is True;, Statement-2 is a correct explanation for, Statement-1, (B) Statement-1, is True, Statement-2 is True;, Statement-2 is NOT a correct explanation for, Statement-1, (C) Statement-1 is True, Statement-2 is False, (D) Statement-1, False, Statement-2 is True, , PHOTON, , INDEFINITE INTEGRATION, , 42

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ANSWER KEY, LEVEL-1, Q.No., , 1, , Ans., , B, , B, , A, , B, , B, , C, , B, , D, , A, , D, , C, , B, , A, , C, , A, , A, , A, , A, , A, , A, , Q.No. 21, , 22, , 23, , 24, , 25, , 26, , 27, , 28, , 29, , 30, , 31, , 32, , 33, , 34, , 35, , 36, , 37, , 38, , 39, , 40, , Ans., , 2, , 3, , 4, , 5, , 6, , 7, , 8, , 9, , 10, , 11, , 12, , 13, , 14, , 15, , 16, , 17, , 18, , 19, , 20, , C, , A, , B, , D, , B, , C, , B, , B, , B, , A, , A, , C, , C, , C, , A, , A, , C, , D, , A, , A, , Q.No. 41, , 42, , 43, , 44, , 45, , 46, , 47, , 48, , 49, , 50, , 51, , 52, , 53, , 54, , 55, , 56, , 57, , 58, , 59, , 60, , D, , D, , A, , C, , B, , C, , A, , C, , A, , A, , A, , C, , B, , C, , C, , C, , B, , A, , A, , B, , Q.No. 61, Ans. A, Q.No. 81, , 62, C, 82, , 63, B, 83, , 64, C, 84, , 65, A, 85, , 66, D, 86, , 67, D, 87, , 68, C, 88, , 69, A, 89, , 70, D, 90, , 71, A, 91, , 72, C, 92, , 73, D, 93, , 74, D, 94, , 75, C, 95, , 76, A, 96, , 77, B, 97, , 78, B, 98, , 79, A, 99, , 80, C, 100, , Ans. D, Q.No. 101, , A, 102, , B, 103, , B, 104, , C, 105, , A, 106, , A, 107, , D, 108, , C, 109, , C, 110, , A, 111, , B, 112, , B, 113, , B, 114, , B, 115, , D, 116, , A, 117, , C, 118, , C, 119, , A, 120, , D, , A, , A, , B, , C, , A, , C, , B, , A, , C, , A, , B, , B, , B, , A, , C, , A, , A, , B, , 122, , 123, , 124, , 125, , 126, , 127, , 128, , 129, , 130, , 131, , 132, , 133, , 134, , 135, , 136, , 137, , 138, , 139, , 140, , C, , C, , A, , A, , A, , C, , C, , D, , B, , A, , B, , B, , A, , C, , C, , B, , A, , D, , A, , Ans., , Ans., , B, , Q.No. 121, Ans., , C, , Q.No. 141, Ans. B, , LEVEL-2, Q.No., , 1, , 2, , 3, , 4, , 5, , 6, , 7, , 8, , 9, , 10, , 11, , 12, , 13, , 14, , 15, , 16, , 17, , 18, , 19, , 20, , Ans., , C, , C, , C, , C, , C, , D, , C, , B, , B, , D, , A, , B, , C, , C, , C, , A, , B, , C, , A, , C, , Q.No., , 21, , 22, , 23, , 24, , 25, , 26, , 27, , 28, , 29, , 30, , 31, , 32, , 33, , 34, , 35, , 36, , 37, , 38, , 39, , 40, , A, , A, , A, , A, , A, , B, , C, , A, , A, , D, , A, , B, , A, , Ans., , A, , B, , A, , A, , D, , C, , A, , Q.No., , 41, , 42, , 43, , 44, , 45, , 46, , 47, , Ans., , C, , B, , C, , C, , B, , C, , D, , LEVEL-3, Q.No., , 1, , 2, , 3, , 4, , 5, , 6, , 7, , 8, , 9, , 10, , 11, , 12, , 13, , 14, , 15, , 16, , 17, , 18, , 19, , 20, , Ans., , D, , C, , C, , A, , D, , D, , D, , D, , A, , D, , C, , C, , A, , C, , B, , D, , D, , A, , B, , B, , Q.No., , 21, , 22, , 23, , 24, , 25, , 26, , 27, , Ans., , D, , A, , C, , A, , A, , A, , C, , LEVEL-4, SECTION-A, Q.No., , 1, , 2, , 3, , 4, , 5, , 6, , Ans., , C, , B, , D, , D, , A, , B, , SECTION-B, , PHOTON, , Q.No., , 1, , 2, , 3, , 4, , 5, , 6, , 7, , Ans., , B, , A, , A, , B, , B, , D, , C, , INDEFINITE INTEGRATION, , 43