Indefinite Integration PYQs Set 3 (PDF)

Set 3 of 16 · Questions 61-90 of 459

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Indefinite Integration · Set 3 (Q61-90)
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61. $\int_{0.2}^{3.5} [x] dx is equal to$ (1) $3.5$ (2) $4.5$ (3) $3$ (4) $4$ 62. $If \int e^{x} (\frac{x s i n ^{- 1} x}{1 - x ^{2}} + \frac{s i n ^{- 1} x}{( 1 - x ^{2} ) ^{3/2}} + \frac{x}{1 - x ^{2}}) d x = g (x) + C, where C is the constant of integration, then g (\frac{1}{2}) equals:$ (1) $\frac{π}{4} \frac{e}{3}$ (2) $\frac{π}{6} \frac{e}{3}$ (3) $\frac{π}{4} \frac{e}{2}$ (4) $\frac{π}{6} \frac{e}{2}$ 63. $If f (x) is a polynomial of the second degree in x such that f (0) = f (1) = 3 f (2) = - 3 . Then \int \frac{f ( x )}{x ^{3} - 1} dx =$ (1) $lo g (\frac{x ^{2} + x + 1}{∣ x - 1∣}) + \frac{1}{3} tan^{- 1} (\frac{2 x + 1}{3}) + c$ (2) $lo g (\frac{x ^{2} + x + 1}{∣ x - 1∣}) - \frac{2}{3} tan^{- 1} (\frac{2 x + 1}{3}) + c$ (3) $lo g (\frac{x ^{2} + x + 1}{∣ x - 1∣}) - \frac{1}{3} tan^{- 1} (\frac{2 x + 1}{3}) + c$ (4) $lo g (\frac{x ^{2} + x + 1}{∣ x - 1∣}) + \frac{2}{3} tan^{- 1} (\frac{2 x + 1}{3}) + c$ 64. $The value of \int e^{x} (x^{5} + 5 x^{4} + 1) dx is$ (1) $e^{x} \cdot x^{5} + C$ (2) $e^{x} \cdot x^{5} + e^{x} + C$ (3) $e^{x + 1} \cdot x^{5} + C$ (4) $5 x^{4} \cdot e^{x} + C$ 65. $Reduction of proper fraction into a sum of partial fraction depends up on factorization of$ (1) $f (x) alone$ (2) $g (x) alone$ (3) $both f (x) and g (x)$ (4) $factors of f (x) and g (x)$ 66. $The value of \int \frac{x e ^{x} dx}{( 1 + x ) ^{2}} is equal to$ (1) $e^{x} (1 + x) + C$ (2) $e^{x} (1 + x^{2}) + C$ (3) $c^{x} (1 + x)^{2} + C$ (4) $\frac{e ^{x}}{1 + x} + C$ 67. $If \int f (x) sin x \cdot cos xdx = \frac{1}{2 ( b ^{2} - a ^{2} )} log f (x) + c_{,} where c is the constant of integration, then f (x) =$ (1) $\frac{2}{( b ^{2} - a ^{2} ) cos 2 x}$ (2) $\frac{2}{ab cos 2 x}$ (3) $\frac{2}{( b ^{2} - a ^{2} ) sin 2 x}$ (4) $\frac{2}{ab sin 2 x}$ 68. $If f (x) = \int \frac{5 x ^{8} + 7 x ^{6}}{( x ^{2} + 2 x ^{7} + 1 ) ^{2}} dx (x \geq 0) and f (0) = 0, then the value of f (1) =$ (1) $\frac{- 1}{2}$ (2) $\frac{- 1}{4}$ (3) $\frac{1}{4}$ (4) $\frac{1}{2}$ 69. $\int x - 1 (x x + 1)^{- 1} d x =$ (1) $ln (x + x^{2} - 1) - sec^{- 1} (x) + c$ (2) $ln (x - x^{2} - 1) - tan^{- 1} (x) + c$ (3) $ln (x + x^{2} - 1) + sec^{- 1} \cdot (x) + c$ (4) $ln (x + x^{2} - 1) - tan^{- 1} (x) + c$ 70. $g (x) is an anti derivative of f (x) = 1 + 2^{x} log 2 and the graph of y = g (x) passes through (- 1, \frac{1}{2}) . Then the curve meets the Y - axis at$ (1) $(0, 1)$ (2) $(0, 2)$ (3) $(0, - 2)$ (4) $(1, 1)$ 71. $The value of \int \frac{e ^{x} ( x ^{2} tan ^{- 1} x + tan ^{- 1} x + 1 )}{x ^{2} + 1} dx is equal to$ (1) $e^{x} tan^{- 1} x + c$ (2) $tan^{- 1} (e^{x}) + c$ (3) $tan^{- 1} (x^{e}) + c$ (4) $e^{tan^{- 1} x} + c$ 72. $The value of \int_{2}^{8} \frac{10 - x}{x + 10 - x} dx is$ (1) $10$ (2) $0$ (3) $8$ (4) $3$ 73. $If \int e^{x} (f (x) - f^{’} (x)) = g (x) + C, then \int e^{x} f^{’} (x) dx =$ (1) $\frac{1}{2} (e^{x} f (x) - g (x)) + C$ (2) $\frac{1}{2} (e^{x} f (x) + g (x)) + C$ (3) $\frac{e ^{x} f ^{’} ( x ) + g ( x )}{2} + C$ (4) $\frac{1}{2} (e^{x} f (x) + e^{x} g (x)) + C$ 74. $\int (\frac{l o g x - 1}{1 + ( l o g x ) ^{2}})^{2} d x =$ (1) $\frac{log x}{1 + ( log x ) ^{2}} + c$ (2) $\frac{x}{x ^{2} + 1} + c$ (3) $\frac{x}{1 + ( log x ) ^{2}} + c$ (4) $\frac{- x}{1 + ( log x ) ^{2}} + c$ 75. $If \int \frac{cos x - sin x}{8 - sin 2 x} dx = \frac{1}{p} log (\frac{3 + sin x + cos x}{3 - sin x - cos x}) + C, then p =$ (1) $6$ (2) $1$ (3) $3$ (4) $12$ 76. $If \int \frac{1 - x ^{2}}{x ^{4}} dx = A (x) (1 - x^{2})^{m} + C for a suitable chosen integer m and a function A (x), where C is a constant of integration, then (A (x))^{m} equals:$ (1) $\frac{- 1}{27 x ^{9}}$ (2) $\frac{- 1}{3 x ^{3}}$ (3) $\frac{1}{27 x ^{6}}$ (4) $\frac{1}{9 x ^{4}}$ 77. $If I_{n} = \int x^{n} sin xdx and I_{6} - 360 I_{2} = f (x) cos x + g (x) sin x, then f (1) + g (1) =$ (1) $-85$ (2) $0$ (3) $-53$ (4) $75$ 78. $\sum \frac{x ^{3} - 1}{x ^{3} + x} dx =$ (1) $x + log \| x \| + \frac{1}{2} log (x^{2} + 1) + sin^{- 1} (x) + c$ (2) $x - log \| x \| + \frac{1}{2} log (x^{2} + 1) - sin^{- 1} (x) + c$ (3) $x + log \| x \| - \frac{1}{2} log (x^{2} + 1) + tan^{- 1} (x) + c$ (4) $x - log \| x \| + \frac{1}{2} log (x^{2} + 1) - tan^{- 1} (x) + c$ 79. $\int_{- 1}^{1} \frac{sin x - x ^{2}}{3 - \| x \|} dx =$ (1) $7 + 18 \cdot log 3/2$ (2) $18 log 9/4$ (3) $7. + 9 log 9/4$ (4) $7 - 18 log 3/2$ 80. $If I (x) = \int x^{2} (log x)^{2} dx and I (1) = 0, then I (x)$ (1) $\frac{x ^{3}}{18} (8 (log x)^{2} - 3 log x) + \frac{7}{18}$ (2) $\frac{x ^{3}}{27} (9 (log x)^{2} + 6 log x) - \frac{2}{27}$ (3) $\frac{x ^{3}}{27} (9 (log x)^{2} - 6 log x + 2) - \frac{2}{27}$ (4) $\frac{x ^{3}}{27} (9 (log x)^{2} - 6 log x - 2) + \frac{2}{27}$ 81. $\int_{0}^{π /2} \frac{1}{a ^{2} sin ^{2} x + b ^{2} cos ^{2} x} dx is equal to$ (1) $\frac{πa}{4 b}$ (2) $\frac{πb}{4 a}$ (3) $\frac{π}{2 ab}$ (4) $\frac{πa}{2 b}$ 82. $\int \frac{2 + x}{2 - x} dx is equal to$ (1) $2 sin^{- 1} (\frac{x}{2}) + 4 - x^{2} + C$ (2) $cos^{- 1} (\frac{x}{2}) - 4 - x^{2} + C$ (3) $sin^{- 1} (\frac{x}{2}) - 4 - x^{2} + C$ (4) $2 sin^{- 1} (\frac{x}{2}) - 4 - x^{2} + C$ 83. $\int \frac{dx}{2 + cos x} = (where C is a constant integration.)$ (1) $\frac{2}{3} tan^{- 1} (\frac{tan ( \frac{x}{2} )}{3}) + C$ (2) $\frac{1}{3} tan^{- 1} (\frac{tan ( \frac{x}{2} )}{3}) + C$ (3) $\frac{1}{2 3} tan^{- 1} (\frac{tan ( \frac{x}{2} )}{2 3}) + C$ (4) $\frac{2}{3} tan^{- 1} (\frac{tan ( \frac{x}{2} )}{2 3}) + C$ 84. $If \int_{0}^{10} f (x) dx = 5, then \sum_{k = 1}^{10} \int_{0}^{1} f (k - 1 + x) dx =$ (1) $50$ (2) $10$ (3) $5$ (4) $20$ 85. $Let \int x^{3} sin x d x = g (x) + C, where C is the constant of integration. If 8 (g (\frac{π}{2}) + g^{'} (\frac{π}{2})) = α π^{3} + β π^{2} + γ, α, β, γ \in Z, then α + β - γ equals:$ (1) $48$ (2) $55$ (3) $62$ (4) $47$ 86. $If \int \frac{2 dx}{cot ^{2} x - tan ^{2} x} = - f (x) + c, then f (x)$ (1) $cot x$ (2) $sin 2 x$ (3) $cos 2 x$ (4) $tan x$ 87. $Find the value of ' k', if it is given that \int_{0}^{b - c} f (x + c) dx = k \int_{c}^{b} f (x) dx$ (1) $1$ (2) $2$ (3) $0$ (4) $-2$ 88. $If \int x^{5} e^{- x^{2}} dx = g (x) e^{- x^{2}} + c Where c is a constant ofintegration, then g (- 1) is equal to:$ (1) $- 1$ (2) $1$ (3) $- \frac{5}{2}$ (4) $- 1 2$ 89. $\int (\frac{x}{a} + \frac{b}{x} + x^{a} + b^{x} + ab) dx is equal to$ (1) $\frac{x ^{2}}{2 a} + \frac{b}{x ^{2}} + \frac{x ^{a + 1}}{a + 1} + \frac{b ^{x}}{log b} + C$ (2) $\frac{x ^{2}}{2 a} + blog \| x \| + \frac{x ^{a + 1}}{a + 1} + \frac{b ^{x}}{log b} + abx + C$ (3) $\frac{1}{a} + blog \| x \| + a x^{a - 1} + b^{x} log b + ab + C$ (4) $\frac{x ^{2}}{2 a} + blog \| x \| + \frac{x ^{a + 1}}{a + 1} + \frac{b ^{x}}{log a} + abx + C$ 90. $If \sum \frac{( 1 + x ^{2} + x ) ^{10}}{( 1 + x ^{2} - x ) ^{9}} dx = \frac{1}{m} ((1 + x^{2} + x)^{n} ((n 1 + x^{2} - x)) + C) where C is the constant of integration and m, n \in N, then m + n is equal to ______.$