Indefinite Integration PYQs Set 2 (PDF)

Set 2 of 16 · Questions 31-60 of 459

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Indefinite Integration · Set 2 (Q31-60)
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31. $\int \frac{x e ^{2 x}}{( 1 + 2 x ) ^{2}} dx = (where C is a constant of integration.)$ (1) $\frac{e ^{2 x}}{1 + 2 x} + C$ (2) $\frac{e ^{2 x}}{2 ( 1 + 2 x )} + C$ (3) $\frac{4 e ^{2 x}}{1 + 2 x} + C$ (4) $\frac{e ^{2 x}}{4 ( 1 + 2 x )} + C$ 32. $If x \neq = (2 n + 1) \frac{π}{2}, n \in Z and cos x \neq = \frac{- 1}{2} then \int (\frac{sin x + sin 2 x}{1 + cos x + cos 2 x})^{2} dx =$ (1) $\frac{tan ^{3} x}{3} - x + c$ (2) $\frac{sec ^{3} x}{3} - x + c$ (3) $cot x - x + c$ (4) $tan x - x + c$ 33. $\int (1 + \frac{x}{1 !} + \frac{x ^{2}}{2 !} + \dots \infty) d x =$ (1) $log (x + 1) + c$ (2) $\frac{1}{x + 1} + c$ (3) $e^{x} + c$ (4) $- e^{- x} + c$ 34. $\int_{0}^{1} \frac{1 + x}{1 - x} dx =$ (1) $\frac{π}{2} - 1$ (2) $\frac{π}{2}$ (3) $\frac{π}{2} + 1$ (4) $\frac{1}{2}$ 35. $\int \frac{e ^{x} ( x + 3 )}{( x + 5 ) ^{3}} dx is equal to$ (1) $\frac{e ^{x}}{( x + 5 ) ^{2}} + C$ (2) $e^{x} (x + 5)^{2} + C$ (3) $e^{x} (x + 3)^{2} + C$ (4) $\frac{e ^{x}}{( x + 3 ) ^{2}} + C$ 36. $If f (x) = \int \frac{2 - 3 sin ^{2} x}{1 + cos 2 x} dx and f (\frac{π}{4}) = 1 then f (0) =$ (1) $\frac{3}{8} (4 - π)$ (2) $3 - \frac{π}{4}$ (3) $0$ (4) $1$ 37. $For x > 0, if \int (log x)^{5} dx is equal to x (A (log x)^{5} +) (B (log x)^{4} + C (log x)^{3} + D (log x)^{2} + E (log x) + F) + constant, then A + B + C + D + E + F is equal to$ (1) $-44$ (2) $-42$ (3) $-40$ (4) $-36$ 38. $If \int_{0}^{a} \frac{dx}{4 + x ^{2}} = \frac{π}{8}, then the value of a is equal to$ (1) $1$ (2) $2$ (3) $3$ (4) $4$ 39. $\sum \frac{( x + 1 ) ^{2}}{x ( x ^{2} + 1 )} dx =$ (1) $log (x (x^{2} + 1)) + c$ (2) $log \| x \| + c$ (3) $log \| x \| + 2 tan^{- 1} (x) + c$ (4) $2 log \| x \| + tan^{- 1} (x) + c$ 40. $\int_{0}^{π /2} sin θ cos^{3} θdθ is equal to$ (1) $\frac{8}{23}$ (2) $\frac{8}{21}$ (3) $\frac{7}{23}$ (4) $\frac{7}{21}$ 41. $If \frac{dθ}{cos ^{2} θ ( tan 2 θ + sec 2 θ )} = λ tan θ + 2 log \| f (θ) \| + C where C is a constant of integration, then the ordered pair (λ, f (θ)) is equal to$ (1) $(1, 1 + tan θ)$ (2) $(1, 1 - tan θ)$ (3) $(- 1, 1 + tan θ)$ (4) $(- 1, 1 - tan θ)$ 42. $If \int x^{3} e^{5 x} dx = \frac{e ^{5 x}}{5 ^{4}} [f (x)] + C, then f (x) is equal to$ (1) $\frac{x ^{3}}{5} - \frac{3 x ^{2}}{5 ^{2}} + \frac{6 x}{5 ^{3}} - \frac{6}{5 ^{4}}$ (2) $5 x^{3} - 5^{2} x^{2} + 5^{3} x - 6$ (3) $5^{2} x^{3} - 15 x^{2} + 30 x - 6$ (4) $5^{3} x^{3} - 75 x^{2} + 30 x - 6$ 43. $Let f (t) = \int (\frac{1 - s i n ( l o g _{e} t )}{1 - c o s ( l o g _{e} t )}) d t, t > 1 . If f (e^{π /2}) = - e^{π /2} and f (e^{π /4}) = α e^{π /4}, then α equals$ (1) $1 + 2$ (2) $- 1 - 2$ (3) $- 1 - 22$ (4) $- 1 + 2$ 44. $If \int \frac{e ^{x}}{x} (x + x) dx = e^{x} [Ax + B x + C] + K, then A + B + C =$ (1) $-2$ (2) $2$ (3) $4$ (4) $-4$ 45. $\int \frac{1 + x}{1 - x} dx = (where C is a constant of integration)$ (1) $sin^{- 1} x + 1 - x^{2} + C$ (2) $1 - x^{2} - x + C$ (3) $- 1 - x^{2} + 1 + x + C$ (4) $sin^{- 1} x - 1 - x^{2} + C$ 46. $\sum \frac{dx}{cos ^{2} ( x ) + sin ( 2 x )} =$ (1) $\frac{1}{2} log \| 1 + 2 cos (x) \| + c$ (2) $\frac{1}{2} log \| 1 - 2 tan (x) \| + c$ (3) $\frac{1}{2} log \| 1 + 2 tan (x) \| + c$ (4) $\frac{1}{2} log \| 1 + 2 cot (x) \| + c$ 47. $If \int \frac{( 2 x + 1 ) ^{6}}{( 3 x + 2 ) ^{8}} dx = P (\frac{2 x + 1}{3 x + 2})^{Q} + R, then \frac{P}{Q} is equal to$ (1) $1 7^{2}$ (2) $\frac{1}{7}$ (3) $7^{2}$ (4) $7$ 48. $Evaluate \int_{2}^{3} x^{2} dx as the limit of a sum$ (1) $\frac{72}{6}$ (2) $\frac{25}{7}$ (3) $\frac{53}{9}$ (4) $\frac{19}{3}$ 49. $If \int \frac{dx}{( x ^{2} - 2 x + 10 ) ^{2}} = A (tan^{- 1} (\frac{x - 1}{3}) + \frac{f ( x )}{x ^{2} - 2 x + 10}) + C Where C is a constant of integration, then:$ (1) $A = \frac{1}{54} and f(x)=3(x-1)$ (2) $A = \frac{1}{81} and f(x)=3(x-1)$ (3) $A = \frac{1}{27} and f (x) = 9 (x - 1)$ (4) $A = \frac{1}{54} and f (x) = 9 (x - 1)^{2}$ 50. $If a, b, c, d \neq = 0 and f (\frac{at + b}{ct + d}) = t, then \int f (x) dx =$ (1) $\frac{- d}{c} x + \frac{bc - ad}{c ^{2}} log (cx - a) + K$ (2) $\frac{- d}{c} x + \frac{bc - ad}{c ^{2}} log (cx + d) + K$ (3) $\frac{a}{c} x + \frac{ad - bc}{c ^{2}} log (cx + a) + K$ (4) $\frac{a}{c} x - \frac{ad - bc}{c} log (cx + a) + K$ 51. $\int \frac{x ^{3} sin ( tan ^{- 1} ( x ^{4} ) )}{1 + x ^{8}} dx is equal to$ (1) $\frac{- cos ( tan ^{- 1} ( x ^{4} ) )}{4} + C$ (2) $\frac{cos ( tan ^{- 1} ( x ^{4} ) )}{4} + C$ (3) $\frac{- cos ( tan ^{- 1} ( x ^{3} ) )}{3} + C$ (4) $\frac{sin ( tan ^{- 1} ( x ^{4} ) )}{4} + C$ 52. $If \int \frac{2 dx}{cos x sin 2 x} = f (x) + c, then f (x) =$ (1) $2 sec x$ (2) $tan x$ (3) $2 tan x$ (4) $22 tan x$ 53. $\int_{0}^{1} \frac{x e ^{x}}{( 2 + x ) ^{3}} dx is equal to$ (1) $\frac{1}{27} \cdot e - \frac{1}{8}$ (2) $\frac{1}{9} \cdot e + \frac{1}{4}$ (3) $\frac{1}{27} \cdot e + \frac{1}{8}$ (4) $\frac{1}{9} \cdot e - \frac{1}{4}$ 54. $\int \frac{1}{x ^{2} ( x ^{4} + 1 ) ^{3/4}} dx is equal to$ (1) $\frac{- ( 1 + x ^{4} ) ^{1/4}}{x ^{2}} + C$ (2) $\frac{- ( 1 + x ^{4} ) ^{3/4}}{x} + C$ (3) $\frac{- ( 1 + x ^{4} ) ^{1/4}}{x} + C$ (4) $\frac{- ( 1 + x ^{4} ) ^{1/4}}{2 x} + C$ 55. $If \frac{x ^{4}}{( x - 1 ) ( x - 2 ) ( x - 3 )} = x + k + \frac{A}{x - 1} + \frac{B}{x - 2} + \frac{C}{x - 3}, then k + A - B + C =$ (1) $104$ (2) $52$ (3) $63$ (4) $\frac{127}{2}$ 56. $\sum \frac{3 sin x + 5 cos x + 4}{sin x + cos x + 2} dx =$ (1) $lo g (sin x + cos x + 2) + 4 x - 4 tan^{- 1} (1 + tan \frac{x}{2}) + c$ (2) $lo g (sin x + cos x + 2) + 4 x - 42 tan^{- 1} (\frac{1 + t a n \frac{x}{2}}{2}) + c$ (3) $4 lo g (sin x + cos x + 2) + x - 42 tan^{- 1} (\frac{1 + t a n \frac{x}{2}}{2}) + c$ (4) $4 lo g (sin x + cos x + 2) + 4 x - 42 tan^{- 1} (\frac{1 - t a n \frac{x}{2}}{2}) + c$ 57. $\sum \frac{2 tan ( x )}{1 + 2 tan ^{2} ( x )} dx =$ (1) $log (cos^{2} x + sin^{2} x) + c$ (2) $log (\frac{cos ^{2} x}{2} + sin^{2} x) + c$ (3) $log (cos^{2} x + \frac{sin ^{2} x}{2}) + c$ (4) $log (\frac{cos ^{2} x}{2} + \frac{sin ^{2} x}{2}) + c$ 58. $If \int \frac{cos x + x}{1 + sin x} dx = f (x) + \int \frac{3 cos \frac{x}{2} - sin \frac{x}{2}}{cos \frac{x}{2} + sin \frac{x}{2}} dx + c then f (x) =$ (1) $\frac{- 2 x}{1 + tan \frac{x}{2}}$ (2) $\frac{- xcos \frac{x}{2}}{cos \frac{x}{2} + sin \frac{x}{2}}$ (3) $\frac{2 x}{1 + tan \frac{x}{2}}$ (4) $\frac{xcos \frac{x}{2}}{cos \frac{x}{2} + sin \frac{x}{2}}$ 59. $3 x 7^{22} + 2 x 1 0^{22} -44when divided by 18 leaves the remainder.$ 60. $If \int cosec^{5} xdx = α cot x cosec x (cossc^{2} x + \frac{3}{2}) + β log_{ϵ} (tan \frac{x}{2}) + C where α, β \in R and C is the constant of integration, then the value of 8 (α + β) equals_______$