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TWO MARK QUESTIONS: Evaluate the following integrals:, Write an antiderivative for each of the, following functions using differentiation :, Question 1:, The anti derivative of, , is the function, , of x whose derivative is, It is known that,, , Question 5:, , ., , Therefore, the anti derivative of, ., Question 2:, The anti derivative of, , is the, , Question 6:, , function of x whose derivative is, ., It is known that,, , Therefore, the anti derivative of, is, , ., , Evaluate the following integrals:, (Question 3 to 14), Question 7:, , Question 3:, , Simplifying and dividing by x-1, we obtain, , Question 4:, , KHV’S, , PAGE 2 http://pue.kar.nic.in

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Question 8:, , Question 12:, , Question 13:, , Question 9:, , Question 14:, , Question 10:, , Question 15:Find the anti derivative of, , Solution:, , Question 11:, , KHV’S, , PAGE 3 http://pue.kar.nic.in

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3 mark questions:, If, , such that f(2) = 0, find f(x), , Also,, , Solution: It is given that, ∴Anti derivative of, , ∴, INTEGRATION BY SUBSTITUTION:, ONE MARK QUESTIONS:, 1`. Evaluate tan 2 (2 x).dx, , , , , ,  tan (2 x).dx   (sec, 2, , 2, , x,  dx ., 2, , 2. Evaluate cos ec , 2, , 2 x  1)dx, , 𝑥, , = -2cot2+c, , Solution:  1 tan 2 x  x  c, , 2, , TWO MARK QUESTIONS:, Integrate the following functions w.r.t x, , Question 1., Hint:, , =t, , Ans: log(1+x2) +c, , Question 4:sin x ⋅ sin (cos x), sin x ⋅ sin (cos x), Let cos x = t, ∴ −sin x dx = dt, , Question 2., Hint: log |x| = t, , ∴, , Question 5:, Let ax + b = t ⇒ adx = dt, , Question 3:, , Let 1 + log x = t, ∴, KHV’S, , PAGE 4 http://pue.kar.nic.in

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Question 10:, , Let, ∴, , Let sin x + cos x = t ⇒ (cos x − sin x) dx = dt, , Question 11:, , ∴, , Let, , Question 14:, , Question 12:, , Let, , ∴, Put cos x − sin x = t ⇒ (−sin x − cos x) dx = dt, , Question 13:, , KHV’S, , PAGE 9 http://pue.kar.nic.in

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Question 4:, , Equating the coefficients of x and, constant term on both sides, we obtain, 4A = 4 ⇒ A = 1, A+B=1⇒B=0, Let 2x2 + x − 3 = t, ∴ (4x + 1) dx = dt, , Question 5:, , Equating the coefficients of x and constant, term on both sides, we obtain, , Question 3:, , From (1), we obtain, , From equation (2), we obtain, , KHV’S, , PAGE 17 http://pue.kar.nic.in

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Question 6: Integrate, , x2, x 2  2x  3, , with respect to x., , I, , 1,  2x  2   1, x2, 2, dx  , dx, x 2  2x  3, x 2  2x  3, , , , 1, 2x  2, 1, dx  , dx, , 2 x 2  2x  3, x 2  2x  3, 1, 1,   2 x 2  2x  3  , dx, 2, 2, x, , 1, , 2,  ,  x 2  2x  3  log  x  1  x 2  2x  3  c, , INTEGRATION BY PARTIAL FRACTIONS, TWO MARK QUESTIONS:, Question 1:, Let, Equating the coefficients of x and constant, term, we obtain, A + B = 1 ; 2A + B = 0, On solving, we obtain A = −1 and B = 2, , THREE MARK QUESTIONS:, Question 1:, Let, , Substituting x = 1, 2, and 3 respectively in, equation (1), we obtain, A = 1, B = −5, and C = 4, , Question 2:, Let, Equating the coefficients of x and constant, term, we obtain, A + B = 0 ; −3A + 3B = 1, On solving, we obtain, , Question 2:, Let, , Substituting x = 1, 2, and 3 respectively in, equation (1), we obtain, , KHV’S, , PAGE 18 http://pue.kar.nic.in

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Equating the coefficients of x2, x, and constant, term, we obtain, A + C = 0 ;−A + B = 1 ; −B + C = 0, On solving these equations, we obtain, Question 3:, From equation (1), we obtain, , Let, Substituting x = −1 and −2 in equation (1), we, obtain, A = −2 and B = 4, , Question 4:, It can be seen that the given integrand is not a, proper fraction. Therefore, on dividing (1 −, x2) by x(1 − 2x), we obtain, , Let, , Question 5:, , Substituting x = 0 and in equation (1), we, obtain A = 2 and B = 3, , Substituting in equation (1), we obtain, , Let, , Substituting x = 1, we obtain, Equating the coefficients of x2 and constant, term, we obtain, A + C = 0 ;−2A + 2B + C = 0, On solving, we obtain, , Question 6:, , Question 4:, Let, KHV’S, , PAGE 19 http://pue.kar.nic.in

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Let, , Substituting x = 1 in equation (1), we obtain, B=4, Equating the coefficients of x2 and x, we, obtain A + C = 0; B − 2C = 3, Question 9:, It can be seen that the given integrand is not a, proper fraction., Therefore, on dividing (x3 + x + 1) by x2 − 1,, , On solving, we obtain, , we obtain, Let, Substituting x = 1 and −1 in equation (1), we, Question 7:, , obtain, , Let, , Equating the coefficients of x2 and x, we, obtain, , Question 10:, , Equating the coefficient of x and constant, term, we obtain, A=3, 2A + B = −1 ⇒ B = −7, , Question 8:, , Let, Question 11:, [Hint: multiply numerator and denominator by, xn − 1 and put xn = t], Substituting x = −1, −2, and 2 respectively in, equation (1), we obtain, KHV’S, , PAGE 20 http://pue.kar.nic.in

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Let x2 = t ⇒ 2x dx = dt, , Multiplying numerator and, denominator by xn − 1, we obtain, , Substituting t = −3 and t = −1 in equation (1),, we obtain, Substituting t = 0, −1 in equation (1), we, obtain, A = 1 and B = −1, , Question 14:, , Question 12:, sin x = t], , [Hint: Put, , Multiplying numerator and denominator by x3,, we obtain, , Let x4 = t ⇒ 4x3dx = dt, , Substituting t = 2 and then t = 1 in equation, (1), we obtain A = 1 and B = −1, Substituting t = 0 and 1 in (1), we obtain, A = −1 and B = 1, , Question 13:, , KHV’S, , PAGE 21 http://pue.kar.nic.in

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Substituting x = 1 and 2 in (1), we obtain, A = −1 and B = 2, , Question 15:, [Hint: Put ex = t], Let ex = t ⇒ ex dx = dt, , Question 17:, Substituting t = 1 and t = 0 in equation (1), we, obtain A = −1 and B = 1, , Equating the coefficients of x2, x, and constant, term, we obtain, A + B = 0; C = 0 ; A = 1, On solving these equations, we obtain, A = 1, B = −1, and C = 0, , Question 16:, , INTEGRATION BY PARTS, TWO MARKS QUESTIONS:, Question 1: x sin x, Let I =, Taking x as first function and sin x as second, function and integrating by parts, we obtain, , Question 3: ∫ 𝑙𝑜𝑔𝑥 𝑑𝑥., , Question 2:, , Given Integral, , Let I =, Taking x as first function and sin 3x as second, function and integrating by parts, we obtain, KHV’S, , =log 𝑥 ∙ ∫ 1 𝑑𝑥 − ∫ 1., , == 𝑥 log 𝑥 − 𝑥 + 𝑐, , PAGE 22 http://pue.kar.nic.in, , 𝑑, 𝑑𝑥, , log 𝑥 𝑑𝑥.

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Question 4: x logx, Let, Taking log x as first function and x as second, function and integrating by parts, we obtain, THREE MARKS QUESTIONS:, Integrate the following w.r.t. x, Question 1:, Let, Taking x2 as first function and ex as second, function and integrating by parts, we obtain, Question 5: x log 2x, Let, Taking log 2x as first function and x as second, function and integrating by parts, we obtain, Again integrating by parts, we obtain, , Question 6: x2 log x, , Question 2:, , Let, Taking log x as first function and x2 as second, function and integrating by parts, we obtain, , Let, Taking, as first function and x as, second function and integrating by parts, we, obtain, , Question 7:, Let, Taking x as first function and sec2x as second, function and integrating by parts, we obtain, , KHV’S, , PAGE 23 http://pue.kar.nic.in

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Taking, as first function and 1 as, second function and integrating by parts, we, obtain, Taking, as first function and, as second function and integrating by parts, we, obtain, , Question 7:, , *******, , Let, , INTEGRAL OF THE FORM ∫ 𝒆𝒙 [𝒇(𝒙) + 𝒇′ (𝒙)]𝒅𝒙, TWO MARK QUESTIONS, x  1  sin x , 2. Evaluate:  e ,  dx .,  1  cos x , Question 1:, , x, x, , 1  2sin cos , , 2, 2 dx, I   ex , , 2 x, , , 2cos, , 2, , , , , , 1, x,   ex ,  tan  dx, x, 2,  2cos 2, , 2, , x, x, 1,   e x  sec 2  tan  dx   e x  f '(x)  f (x)  dx, 2, 2, 2, x,  e x tan, 2, , Let, Let, ⇒, ∴, It is known that,, , Question 2:, THREE MARK QUESTIONS, Integrate the following w.r.t x, Also, let, It is known that,, , ⇒, Question 1:, , Let, , KHV’S, , PAGE 25 http://pue.kar.nic.in

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Let, , ⇒, , Let, ⇒, It is known that,, , It is known that,, , Question 3:, , Question 2:, , Let, Integrating by parts, we obtain, , Again integrating by parts, we obtain, , Let, ⇒, It is known that,, , Question 4:, Let, , ⇒, , From equation (1), we obtain, = 2θ, ⇒, Question 3:, KHV’S, , PAGE 26 http://pue.kar.nic.in

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Detail of the concepts to be mastered by every student of class second PUC, with exercises and examples of NCERT Text Book., Indefinit, e, Integrals, , (i) Integration by substitution, , *, , Text book , Vol. II Examples 5&6, Page 300, 302,301,303, , (ii) ) Application of trigonometric, function in integrals, , **, , Text book , Vol. II Ex 7 Page 306,, Exercise 7.3 Q13&Q24, , (iii) Integration of some particular, function, , ***, , Text book , Vol. II Exp 8, 9, 10, Page 311,312,313, Exercise 7.4 Q, 3,4,8,9,13&23, , (iv) Integration using Partial Fraction, , ***, , Text book , Vol. II Exp 11&12, Page 318 Exp 13 319,Exp 14 & 15, Page320, , (v) Integration by Parts, , **, , Text book , Vol. II Exp 18,19&20, Page 325 Exs 7.6 QNO ,10,11,, 17,18,20, , (vi)Some Special Integrals, , ***, , Text book , Vol. II Exp 23 &24, Page 329, , **, , Text book , Vol. II Solved Ex. 40,, 41, ,  x2  a2 , dx, , , , , , , , ,, , 1, a2  x2, , dx, x a, , dx ,, , dx, ax 2  bx  c, (px  q)dx, , 2, , 2, , ,, , dx, ,  ax 2  bx  c ,, ,, , (px  q)dx, ,  ax 2  bx  c ,, , ax 2  bx  c, , a 2  x 2 dx ,, , , , x 2  a 2 dx, , (vii) Miscellaneous Questions, , viii)Some special integrals, , Text book Supplimentary material, Page 614,615, , SYMBOLS USED :, , * : Important Questions, ** :Very Important Questions, *** : Very-Very Important Questions, , KHV’S, , PAGE 30 http://pue.kar.nic.in