Definite Integration PYQs Set 2 (PDF)

Set 2 of 9 · Questions 31-60 of 248

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Definite Integration · Set 2 (Q31-60)
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31. $The value of \int_{- 3}^{3} (a x^{5} + b x^{3} + cx + k) dx, where a, b, c, k are constants, depends only on$ (1) $a, b and c$ (2) $k$ (3) $a and b$ (4) $a and k$ 32. $If \int_{0}^{π /2} \frac{c o t x}{c o t x + cosec x} d x = m (π + n), then m \cdot n is equal to :$ (1) $- \frac{1}{2}$ (2) $1$ (3) $\frac{1}{2}$ (4) $- 1$ 33. $\int_{a}^{b} \frac{x}{x + a + b - x} dx =$ (1) $a + b$ (2) $\frac{b - a}{2}$ (3) $a - b$ (4) $\frac{a - b}{2}$ 34. $The value of the integral \int_{- π /4}^{π /4} \frac{x + \frac{π}{4}}{2 - c o s 2 x} d x is$ (1) $\frac{π ^{2}}{6}$ (2) $\frac{π ^{2}}{12 3}$ (3) $\frac{π ^{2}}{3 3}$ (4) $\frac{π ^{2}}{6 3}$ 35. $Let I = \int_{1}^{2} \frac{dx}{2 x ^{3} - 9 x ^{2} + 12 x + 4} then$ (1) $\frac{1}{9} < I^{2} < \frac{1}{8}$ (2) $\frac{1}{3} < I^{2} < \frac{1}{2}$ (3) $\frac{1}{9} < I < \frac{1}{8}$ (4) $\frac{1}{3} < I < \frac{1}{2}$ 36. $Let \int_{α}^{log_{e} 4} \frac{d x}{e ^{x} - 1} = \frac{π}{6} . Then e^{α} and e^{- α} are the roots of the equation:$ (1) $x^{2} + 2 x - 8 = 0$ (2) $x^{2} - 2 x - 8 = 0$ (3) $2 x^{2} - 5 x + 2 = 0$ (4) $2 x^{2} - 5 x - 2 = 0$ 37. $\int_{- π}^{π} ∣ π - ∣ x ∣ ∣ d x is equal to :$ (1) $π^{2}$ (2) $2 π^{2}$ (3) $2 π^{2}$ (4) $\frac{π ^{2}}{2}$ 38. $If the value of the integral \int_{- 1}^{1} \frac{cos αx}{1 + 3 ^{x}} dx is \frac{2}{π} . Then, a value of α is$ (1) $\frac{π}{3}$ (2) $\frac{π}{6}$ (3) $\frac{π}{4}$ (4) $\frac{π}{2}$ 39. $Evaluate : \int_{0}^{1} \frac{x tan ^{- 1} x}{( 1 + x ^{2} ) ^{3/2}} dx$ (1) $\frac{4 - π}{2 2}$ (2) $\frac{4 + π}{2 2}$ (3) $\frac{4 - π}{4 2}$ (4) $None of these$ 40. $For 0 < a < 1, the value of the integral \int_{0}^{π} \frac{dx}{1 - 2 a cos x + a ^{2}} is :$ (1) $\frac{π ^{2}}{π + a ^{2}}$ (2) $\frac{π ^{2}}{π - a ^{2}}$ (3) $\frac{π}{1 - a ^{2}}$ (4) $\frac{π}{1 + a ^{2}}$ 41. $\int_{- 1}^{1} log (\frac{2 - x}{2 + x}) dx =$ (1) $2$ (2) $0$ (3) $-1$ (4) $1$ 42. $The integral \int_{π /6}^{π /3} sec^{\frac{2}{3}} x cosec^{\frac{4}{3}} x d x is equal to:$ (1) $3^{\frac{5}{6}} - 3^{\frac{2}{3}}$ (2) $3^{\frac{4}{3}} - 3^{\frac{1}{3}}$ (3) $3^{\frac{7}{6}} - 3^{\frac{5}{6}}$ (4) $3^{\frac{5}{3}} - 3^{\frac{1}{3}}$ 43. $Let for f (x) = 7 tan^{8} x + 7 tan^{6} x - 3 tan^{4} x - 3 tan^{2} x, I_{1} = \sum f (x) d x and I_{2} \sum xf (x) d x . Then 7 I_{1} + 12 I_{2} is equal to:$ (1) $2$ (2) $1$ (3) $2 π$ (4) $π$ 44. $If f (x) = a + bx + c x^{2} Where a, b, c \in R then \int_{0}^{1} f (x) dx is$ (1) $\frac{1}{3} (f (1) + f (0) + 2 f (\frac{1}{2}))$ (2) $\frac{1}{6} (f (1) + f (0) + 4 f (\frac{1}{2}))$ (3) $\frac{1}{6} (f (1) + f (0) - 4 f (\frac{1}{2}))$ (4) $\frac{1}{6} (f (1) - f (0) - 4 f (\frac{1}{2}))$ 45. $The value of \int_{0}^{1} (2 x^{3} - 3 x^{2} - x + 1)^{\frac{1}{3}} dx is equal to:$ (1) $0$ (2) $1$ (3) $2$ (4) $-1$ 46. $Let f (x) + 2 f (\frac{1}{x}) = x^{2} + 5 and 2 g (x) - 3 g (\frac{1}{2}) = x, x > 0 . If α = \int_{1}^{2} f (x) dx, and β = \int_{1}^{2} g (x) dx, then the value of 9 α + β is:$ (1) $0$ (2) $10$ (3) $1$ (4) $11$ 47. $The value of k \in N for which the integral I_{n} = \int_{0}^{1} (1 - x^{k})^{n} dx, n \in N, satisfies 147 I_{20} = 148 I_{21} is$ (1) $14$ (2) $8$ (3) $10$ (4) $7$ 48. $The integral \int_{π /6}^{π /3} tan^{3} x \cdot sin^{2} 3 x (2 sec^{2} x \cdot sin^{2} 3 x + 3 tan x \cdot sin 6 x) dx is equal to:$ (1) $\frac{7}{18}$ (2) $\frac{9}{2}$ (3) $- \frac{1}{18}$ (4) $- \frac{1}{9}$ 49. $The value of the integral \int_{0}^{1} x cot^{- 1} (1 - x^{2} + x^{4}) d x is:$ (1) $\frac{π}{2} - \frac{1}{2} log_{e} 2$ (2) $\frac{π}{4} - log_{e} 2$ (3) $\frac{π}{2} - log_{e} 2$ (4) $\frac{π}{4} - \frac{1}{2} log_{e} 2$ 50. $If f^{’} (x) = tan^{- 1} (sec x + tan x), x \in (\frac{- π}{2}, \frac{π}{2}) and f (0) = 0 then the value of f (1) is$ (1) $\frac{π + 1}{4}$ (2) $\frac{π - 1}{4}$ (3) $\frac{π + 1}{2}$ (4) $0$ 51. $Let f (x) = x + \frac{a}{π ^{2} - 4} sin x + \frac{b}{π ^{2} - 4} cos x x ϵ R be a function which satisfies f (x) = x + \sum sin (x + y) f (y) dy . Then (a + b) is equal to$ (1) $- π (π + 2)$ (2) $- 2 π (π + 2)$ (3) $- 2 π (π - 2)$ (4) $- π (π - 2)$ 52. $\int_{0}^{π /2} \frac{sin x}{1 + cos ^{2} x} dx has the value$ (1) $0$ (2) $π /2$ (3) $π /4$ (4) $- π /4$ 53. $If \int_{0}^{a} \frac{a - x}{x} dx = \frac{K}{2}, then K =$ (1) $πa /2$ (2) $5 πa /2$ (3) $3 πa /2$ (4) $πa$ 54. $lim n \to \infty (\frac{1}{1 + n} + \frac{1}{2 + n} + \frac{1}{3 + n} + \dots + \frac{1}{2 n}) is equal to$ (1) $0$ (2) $log_{e} 2$ (3) $log_{3} (\frac{3}{2})$ (4) $log_{e} 2 3$ 55. $The value of the integral \int_{0}^{π /2} log \| tan x + cot x \| dx$ (1) $π log 2$ (2) $- π log 2$ (3) $π /2 log 2$ (4) $None of these$ 56. $If f (x) satisfies the relation f (x) = e^{x} + \sum (y + x e^{x}) f (y) dy, then e + f (0) is equal to____.$ 57. $Let f : (0, \infty) \to R and F (x) = \int_{0}^{x} tf (t) dt . If F (x^{2}) = x^{4} + x^{5}, then \sum_{r = 1}^{12} f (r^{2}) is equal to :$ 58. $If \int_{1/3}^{3} ∣ lo g_{e} x ∣ d x = \frac{m}{n} lo g_{e} (\frac{n ^{2}}{e}), where m and n are coprime natural numbers, then m^{2} + n^{2} - 5 is equal to ______$ 59. $If \int_{0}^{\frac{π}{4}} \frac{sin ^{2} x}{1 + sin x cos x} d x = \frac{1}{a} log_{e} (\frac{a}{3}) + \frac{π}{b 3}, where a, b \in N, then a + b is equal to _______$ 60. $(\frac{120}{π ^{3}} \int_{0}^{π} \frac{x ^{2} sin x cos x}{sin ^{4} x + cos ^{4} x} dx) is equal to________$