Page 1 :
INDEFINITE INTEGRATION, , INDEFINITE INTEGRATION, Integration is the inverse process of differentiation. That, is, the process of finding a function, whose differential, coefficient is known, is called integration., , 3., , ³ k f (x)dx, , 4., , If f1(x), f2(x), f3(x), ... (finite in number) are functions, of x, then, , If the differential coefficient of F(x) is f (x),, i.e., , d, [F(x)], dx, , Here, , ³ dx, , ³ [ f (x) r f, , f (x), then we say that the antiderivative, , or integral of f (x) is F(x), written as ³ f (x)dx, , 1, , 5., , d, [F(x) C], dx, ? ³ f (x)dx, , 2, , (x)dx r ³ f 3 (x)dx r ..., , F(x) c, 1, F(ax r b) c, a, , 1.2 Standard Formulae of Integration, , 1. INDEFINITE INTEGRAL, , The following results are a direct consequence of the, , d, [F(x)], dx, , any, , If ³ f (x)dx, , then ³ f (ax r b)dx, , the integration with respect to x., , for, , (x) r f 3 (x)...]dx, , 1, , integrand, x is the variable of integration and dx denotes, , Also,, , 2, , ³ f (x)dx r ³ f, , F(x),, , is the notation of integration f (x) is the, , We know that if, , k ³ f (x) dx , where k is any constant, , f (x) , then ³ f (x)dx, , arbitrary, , d, [F(x)] 0, dx, , constant, , F(x)., , C,, , definition of an integral., x n 1, C, n z 1., n 1, , 1., , n, ³ x dx, , 2., , ³ x dx, , log | x | C, , 3., , ³e, , ex C, , 4., , x, ³ a dx, , 5., , ³ sin x dx, , cos x C, , 6., , ³ cos x dx, , sin x C, , 7., , ³ sec, , 8., , ³ cos ec x dx, , f (x)., 1, , F(x) C,, , This shows that F(x) and F(x) + C are both integrals of the, , x, , dx, , same function f(x). Thus, for different values of C, we obtain, different integrals of f(x). This implies that the integral of, f(x) is not definite. By virtue of this property F(x) is called, , ax, C., log e a, , the indefinite integral of f(x)., 1.1 Properties of Indefinite Integration, , 1., , 2., , d ª, f (x)dx º¼, dx ¬ ³, , f (x), , d, , ³ f '(x)dx ³ dx [ f (x)]dx, , f (x) + c, , 2, , x dx, 2, , tan x C, cot x C
Page 2 :
INDEFINITE INTEGRATION, 9., , ³ sec x tan x dx, , 10., , ³ cos ec x cot x dx, , sec x C, , cos ec x C., , 11., , ³ tan x dx, , log | cos x | C, , 12., , ³ cot x dx, , log | sin x | C, , 13., , ³ sec x dx, , log | sec x tan x | C, , 14., , ³ cos ec x dx, , (ii), , When the integrand is of the form xn – 1 f ´(xn), we, put xn = t and nxn – 1 dx = dt., , log | sec x | C., , log | cos ec x cot x | C, , f (ax b), c, a, , f (t), a, , 1, f ´(t) dt, a³, , Thus,, , ³x, , n 1, , f ´(x n )dx, , 1, f (t), n, , ³ f ´(t) dt, , dt, , 1, n, , ³ f ´(t) n, , 1, f (x n ) c, n, , When the integrand is of the form [ f (x)]n . f ´(x), we, , (iii), , put f (x) = t and f ´(x) dx = dt., 15., , ³, , dx, 1 x, , sin 1 x C ; | x | 1, , 2, , Thus,, 16., , dx, , ³ 1 x2, , ³ [ f (x)], , ³x, , dx, x 1, 2, , f ´(x) dx, , ³t, , n, , dt, , t n 1, n 1, , sec1 | x | C ; | x | ! 1, , 2. METHODS OF INTEGRATION, , When the integrand is of the form, , Thus, ³, , f ´(x), dx, f (x), , ³, , dt, t, , log|t| log| f (x)| c, , 2.1.1 Some Special Integrals, By suitable substitution, the variable x in ³ f (x) dx is, , changed into another variable t so that the integrand f (x), , dx, , 1, x, tan 1 C, a, a, , dx, a2, , 1, x a, log, C, 2a, xa, , dx, x2, , 1, ax, log, C, 2a, ax, , 1., , ³ x2 a2, , 2., , ³x, , 3., , ³a, , 4., , ³, , 5., , ³, , is changed into F(t) which is some standard integral or, algebraic sum of standard integrals., There is no general rule for finding a proper substitution, , 2, , and the best guide in this matter is experience., However, the following suggestions will prove useful., If the integrand is of the form f ´ (ax + b), then we, put ax + b = t and dx =, , Thus,, , 1, dt., a, , 2, , dx, a x, 2, , 2, , sin 1, , x, C, a, , dt, , ³ f ´(ax b) dx ³ f ´(t) a, , f ´(x), , we put, f (x), , f (x) = t and f ´(x) dx = dt., , 2.1 Method of Substitution, , (i), , [ f (x)]n 1, c, n 1, , tan 1 x C, , (iv), 17., , n, , dx, x a, 2, , 2, , log x x 2 a 2 C
Page 3 :
INDEFINITE INTEGRATION, , 6., , ³, , 7., , ³, , a x dx, 2, , 2, , x 2 a 2 dx, , 2, , x 2, a2, x, a x 2 sin 1 C, 2, 2, a, , (b), , (c), , ³f, , Make the coefficient of x 2 unity by taking the, coefficient of x2 outside the quadratic., , (ii), , Complete the square in the terms involving x, i.e., write ax 2 + bx + c in the form, 2, 2, ª§, b · º b 4ac, a «¨ x ¸ » , ., 2a ¹ »¼, 4a, «¬©, , x, a2, x 2 a 2 log x x 2 a 2 C, 2, 2, , x 2 a 2 dx, , x, a, x 2 a 2 log x x 2 a 2 C, 2, 2, , 2.1.2 Integrals of the Form, (a), , (i), , (iii), , The integrand is converted to one of the nine special, integrals., , (iv), , Integrate the function., , 2, , ³, , 9., , log x x a C, 2, , x2 a2, , ³, , 8., , Working Role, , dx, , 2.1.4 Integrals of the Form, , a 2 x 2 dx,, , ³f, , px q, , (a), , ³ ax 2 bx c dx,, , (c), , ³ (px q), , (b), , ³, , px q, ax 2 bx c, , dx,, , a 2 x 2 dx,, , ³f, , x 2 a 2 dx,, , ax 2 bx c dx, , Integral Working Rule, px q, , ³ ax 2 bx c dx, , §ax·, (d) ³ f ¨, ¸ dx,, ©ax¹, , Put px + q = O (2ax + b) + P or px + q = O, , (derivative of quadratic) + P., Comparing the coefficient of x and constant term on both, sides, we get, , Working Rule, Integral, , Substitution, , ³f, , a 2 x 2 dx,, , x = a sin T or x = a cos T, , ³f, , a 2 x 2 dx,, , x = a tan T or x = a cot T, , ³f, , x a, , p = 2aO and q = bO + P O =, , 2a, , bp ·, §, and P = ¨ q ¸ .Then, 2a ¹, ©, , integral becomes, px q, , 2, , 2, , x = a sec T or x = a cosec T, , dx,, , §a x·, ³ f ¨© a x ¸¹ dx or, , §a x ·, ³ f ©¨ a x ¸¹ dx, , x, , ³ ax 2 bx c dx, 2ax b, bp ·, dx, §, dx ¨ q ¸ ³ 2, 2a ³ ax 2 bx c, 2a, ax, , bx c, ©, ¹, , a cos 2T, , 2.1.3 Integrals of the Form, , (a) ³, , dx, ,, ax bx c, 2, , (b) ³, , (c) ³ ax 2 bx c dx a ,, , 2a, , dx, ax bx c, 2, , ,, , ³, , bp ·, dx, §, log | ax 2 bx c | ¨ q ¸ ³ 2, 2a ¹ ax bx c, ©, px q, , ax 2 bx c, , dx In this case the integral becomes
Page 4 :
INDEFINITE INTEGRATION, , px q, , ³, , ax bx c, 2, , p, 2a, , ³, , 2.1.6 Integrals of the Form, , dx, , x2 1, ³ x 4 kx 2 1 dx, , 2ax b, , bp ·, dx, §, dx ¨ q ¸ ³, 2, 2a, ©, ¹ ax bx c, ax bx c, , or, , x2 1, ³ x 4 kx 2 1 dx,, , 2, , where k is a constant positive, negative or zero., Working Rule, , p, bp ·, dx, §, ax 2 bx c ¨ q ¸ ³, 2, a, 2a, ©, ¹ ax bx c, , ³ (px q), , ax 2 bx c dx, , (i), , Divide the numerator and denominator by x2., , (ii), , Put x , , 1, x, , z or x , , 1, x, , z , whichever subsitution, on, , differentiation gives, the numerator of the resulting, , The integral in this case is converted to, , integrand., , ³ (px q), , ax 2 bx c dx, , p, 2a, , ³, , 2ax b, , ax 2 bx c dx, , (iii) Evaluate the resulting integral in z, (iv) Express the result in terms of x., , bp ·, §, ¨ q ¸ ³ ax 2 bx c dx, 2a ¹, ©, , 2.1.7 Integrals of the Form, , dx, , p, bp ·, §, (ax 2 bx c)3 / 2 ¨ q ¸ ³ ax 2 bx c dx, 3a, 2a ¹, ©, 2.1.5 Integrals of the Form, , ³, , P(x), ax 2 bx c, , ³P, , Q, , , where P, Q are linear or quadratic functions of x., , Integral, , Substitution, 1, , dx, where P(x) is a polynomial in x of, , degree n t 2., , ³ (ax b), , cx d, , dx, , cx + d = z2, , px q, , px + q = z2, , dx, , ³ (ax 2 bx c), , Working Rule: Write, , ³, , P(x), ax 2 bx c, , ³ (px q), , dx, , (a 0 a1 x a 2 x 2 ... a n 1 x n 1 ), , ax 2 bx c k ³, , dx, ax bx c, 2, , where k, a0, a1, ... an – 1 are constants to be determined by, differentiating the above relation and equating the, coefficients of various powers of x on both sides., , ³ (ax, , dx, ax bx c, 2, , px + q =, , dx, 2, , b) cx d, 2, , x, , 1, z, , 1, z
Page 5 :
INDEFINITE INTEGRATION, quadratic factor which cannot be factorised further:, , 3. METHOD OF PARTIAL FRACTIONS, FOR RATIONAL FUNCTIONS, , g (x) = (ax2 + bx + c) (x – D3) (x – D4) ... (x – Dn)., In such a case express f (x) and g (x) as:, , p (x), can be integrated by resolving, Integrals of the type ³, g (x), , f (x), g (x), , the integrand into partial fractions. We proceed as follows:, Check degree of p (x) and g (x)., If degree of p (x) > degree of g (x), then divide p (x) by g, (x), , till its degree is less, i.e., , put in the, , A3, A1x A 2, An, , ... , 2, x Dn, ax bx c x D 3, , where A1, A2, ... An are constants to be determined by, comparing the coefficients of various powers of x on both, sides after taking L.C.M., CASE 4 : When the denominator contains a repeated, , p (x), form, g (x), , f (x), where degree of f (x) < degree of, r(x) , g (x), , quadratic factor which cannot be factorised further: That, is, g (x) = (ax2 + bx + c)2 (x – D5) (x – D6) ... (x – Dn), , g (x)., CASE 1: When the denominator contains non-repeated, , In such a case write f (x) and g (x) as, , linear factors. That is, g (x) = (x – D) (x – D2) ... (x – Dn)., , f (x), g (x), , A3 x A 4, A1x A 2, , , ax 2 bx c (ax 2 bx c) 2, , In such a case write f (x) and g (x) as:, , f (x), g (x), , A1, A2, An, , ... , (x D1 ) (x D 2 ), (x D n ), , A5, An, ... , x D5, (x D n ), , where A1, A2, ... An are constants to be determined by, , where A1, A2, ... An are constants to be determined by, , comparing the coefficients of various powers of x on both, , comparing the coefficients of various powers of x on both, , sides after taking L.C.M., , sides after taking L.C.M., , CASE 2 : When the denominator contains repeated as well, , CASE 5 : If the integrand contains only even powers of x, , as non-repeated linear factor. That is, , (i), , Put x2 = z in the integrand., , (ii), , Resolve the resulting rational expression in z into, , g (x) = (x – D) (x – D3) ... (x – Dn)., 2, , In such a case write f (x) and g (x) as:, f (x), g (x), , A3, A1, A2, An, , , ... , x D1 (x D1 ) 2 x D 3, (x D n ), , where A 1, A 2 , ... A n are constants to determined by, , partial fractions, (iii), , Put z = x2 again in the partial fractions and then, integrate both sides., , 4. METHOD OF INTEGRATION BY PARTS, , comparing the coefficients of various powers of x on both, sides after taking L.C.M., , The process of integration of the product of two functions is, known as integration by parts., , Note : Corresponding to repeated linear factor (x – a)r in, , For example, if u and v are two functions of x,, , the denominator, a sum of r partial, fractions of the type, , A1, A2, Ar, , ... , is taken., 2, x a (x a), (x a) r, , then ³ (uv ) dx, , § du, ·, u.³ v dx ³ ¨ .³ v dx ¸ dx., © dx, ¹, , In words, integral of the product of two functions = first, function × integral of the second – integral of (differential, , CASE 3 : When the denominator contains a non repeated, , of first × integral of the second function).
Page 6 :
INDEFINITE INTEGRATION, , Working Hints, (i), , (ii), , 4.2 Integrals of the Form:, , Choose the first and second function in such a way that, derivative of the first function and the integral of the second, function can be easily found., In case of integrals of the form ³ f (x). x, , n, , Where the initial integrand reappears after integrating by, parts., Working Rule, , dx, take xn as, , the first function and f (x) as the second function., (iii) In case of integrals of the form ³ (log x ) n 1dx , take 1 as, , (ii), , On integrating by parts second time, we will obtain, , Transpose and collect terms involving I on one side, , (iii), , (iv) Rule of integration by parts may be used repeatedly, if, required., If the two functions are of different type, we can choose, the first function as the one whose initial comes first in the, word “ILATE”, where, , Apply the method of integration by parts twice., , the given integrand again, put it equal to I., , the second function and (log x)n as the first function., , (v), , (i), , and evaluate I., , 5. INTEGRAL OF THE FORM, (TRIGONOMETRIC FORMATS), , I — Inverse Trigonometric function, L — Logarithmic function, , 5.1 (a), , A — Algebraic function, , dx, , ³ a b cos x, , (b) ³, , dx, a b sin x, , T — Trigonometric function, E — Exponential function., , (c), , (vi) In case, both the functions are trigonometric, take that, function as second function whose integral is simple. If, both the functions are algebraic, take that function as first, function whose derivative is simpler., (vii) If the integral consists of an inverse trigonometric function, of an algebraic expression in x, first simplify the integrand, by a suitable trigonometric substitution and then integrate, the new integrand., , dx, , ³ a b cos x c sin x, , Working Rule, , x, x, 2 tan, 2 so that the given, 2, and sin x =, Put cos x =, 2 x, 2 x, 1 tan, 1 tan, 2, 2, , 1 tan 2, , (i), , integrand becomes a function of tan, , x, ., 2, , 4.1 Integrals of the Form, (ii), , x, ³ e ª¬ f x f ´ x ¼º dx, , Put tan, , x, 1, 2 x, = z sec dx, 2, 2, 2, , dz, , (iii) Integrate the resulting rational algebraic function of z, , Working Rule, (i), , Split the integral into two integrals., , (ii), , Integrate only the first integral by parts, i.e., , (iv) In the answer, put z = tan, , x, ., 2, , 5.2 Integrals of the Form, , ³e, , x, , ¬ª f x f ´ x ¼º x, , ³e, , x, , f (x) x ³ e f ´(x) dx, x, , (a) ³, ª f (x).e x f ´(x).e x x º e x f ´(x) dx, ³, ¬, ¼ ³, , e x f (x) C., , dx, a b cos 2 x, , (c) ³, , (b) ³, , dx, a b sin 2 x, , dx, a cos x b sin x cos x c sin 2 x, 2
Page 7 :
INDEFINITE INTEGRATION, Working Rule, (i), , Divide the numerator and denominator by cos2x., , (ii), , In the denominator, replace sec2x, if any, by 1 + tan2x., , (iii) Put tan x = z sec2x dx = dz., , n, , (iv) Integrate the resulting rational algebraic function of z., (v), , In the answer, put z = tan x., , 5.5 Integrals of the Form, , ³ sin, , Working Role, Put Numerator = O (denominator) + P (derivative of, denominator), a cos x + b sin x = O (c cos x + d sin x) + P (– c sin x + d cos x)., Equate coefficients of sin x and cos x on both sides and, find the values of O and P., , (iii) Split the given integral into two integrals and evaluate each, integral separately, i.e., a cos x b sin x, , ³ c cos x d sin x, , m, , x cos n x dx, , Working Rule, (i), , If the power of sin x is an odd positive integer, put cos x = t., , (ii), , If the power of cos x is an odd positive integer, put sin x = t., , (iii) If the power of sin x and cos x are both odd positive integers,, put sin x = t or cos x = t., (iv) If the power of sin x and cos x are both even positive, integers, use De’ Moivre’s theorem as follows:, cos x + i sin x = z. Then cos x – isin x = z–1, , Let,, , dx, , c sin x d cos x, dx, O ³ 1dx P ³, a cos x b sin x, , Adding these, we get z , , Ox P log | a cos x b sin x | ., , By, , (iv) Substitute the values of O and P found in step 2., , a b cos x csin x, ³ e f cos x g sin x dx, , 1, zn, , 2 cos nx and z n , , ?, , n, , sin x cos x, , 1, , Put Numerator = l (denominator) + m, , 2m n, , (derivative of denominator) + n, , Equate coefficients of sin x, cos x and constant term on both, sides and find the values of l, m, n., , a b cos x csin x, , ³ e f cos x g sin x dx, , theorem,, 1, zn, , , , 1, im, , 1, z, , we, , have, , 2i sin n x ...(1), , 1, 1 §, 1· §, 1·, n ¨z ¸ ¨z ¸, m, z¹ ©, z¹, (2i) 2 ©, n, , 2i sin x, , m, , m, , 1· §, 1·, §, ¨z ¸ ¨z ¸ ., z¹ ©, z¹, ©, , Now expand each of the factors on the R.H.S. using, Binomial theorm. Then group the terms equidistant from, the beginning and the end. Thus express all such pairs as, the sines or cosines of multiple angles. Further integrate, term by term., , a + b cos x + c sin x = l (e + f cos x + g sin x) + m, ( – f sin x + g cos x) + n, , (iii) Split the given integral into three integrals and evaluate, each integral separately, i.e., , 2 cos x and z , , n, , m, , Working Rule, , (ii), , 1, z, , De’Moivre’s, , zn , , 5.4 Integrals of the Form, , (i), , dx, dx, e f cos x g sin x, , (iv) Substitute the values of l, m, n found in Step (ii)., , a cos x b sin x, ³ c cos x d sin x dx, , (ii), , dx, , ³ e f cos x g sin x, lx m log | e f cos x g sin x | n ³, , 5.3 Integrals of the Form, , (i), , f sin x g cos x, dx , e f cos x g sin x, , l ³ 1dx m ³, , (v), , If the sum of powers of sin x and cos x is an even negative, integer, put tan x = z.
