Indefinite Integration PYQs Set 4 (PDF)

Set 4 of 16 · Questions 91-120 of 459

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Indefinite Integration · Set 4 (Q91-120)
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91. $\sum sin^{3} (x) \cdot cos^{3} (x) dx =$ (1) $sin^{4} (x) - sin^{6} (x) + c$ (2) $cos^{4} (x) - cos^{6} (x) + c$ (3) $\frac{1}{4} sin^{4} (x) - \frac{1}{6} sin^{6} (x) + c$ (4) $\frac{1}{4} cos^{4} (x) - \frac{1}{6} cos^{6} (x) + c$ 92. $The value of x that satisfies the equation \int_{2}^{x} \frac{dt}{\| t \| t ^{2} - 1} = \frac{π}{12} is$ (1) $1$ (2) $0$ (3) $- 2$ (4) $2$ 93. $If the integral \int \frac{5 tan x}{tan x - 2} dx = x + a ln \| sin x - 2 cos x \| + k, then a is equal to$ (1) $1$ (2) $2$ (3) $-1$ (4) $-2$ 94. $\int (x + 1) (x + 2)^{4} (x + 3) dx is equal to$ (1) $\frac{( x + 1 ) ^{2}}{2} + \frac{( x + 2 ) ^{2}}{5} + \frac{( x + 3 ) ^{2}}{2} + C$ (2) $\frac{( x + 2 ) ^{7}}{7} - \frac{( x + 2 ) ^{5}}{5} + C$ (3) $\frac{( x + 2 ) ^{7}}{7} + \frac{( x + 2 ) ^{5}}{5} + C$ (4) $\frac{( x + 3 ) ^{7}}{7} - \frac{( x + 3 ) ^{5}}{5} + C$ 95. $Let I (x) = \sum \frac{3 dx}{( 4 x + 6 ) ( 4 x ^{2} + 8 x + 3 )} and I (0) = \frac{3}{4} + 20 . If I (\frac{1}{2}) = \frac{a 2}{b} + c, where a, b, c \in N, gcd (a, b) = 1, then a + b + c is equal to$ (1) $30$ (2) $31$ (3) $29$ (4) $28$ 96. $If \int \frac{2 x ^{12} + 5 x ^{9}}{( 1 + x ^{3} + x ^{5} ) ^{3}} dx = \frac{x ^{m}}{l ( 1 + x ^{3} + x ^{5} ) ^{r}} + C then \frac{m - l}{r} =$ (1) $3$ (2) $4$ (3) $5$ (4) $6$ 97. $The value of \int \frac{1 + x ^{4}}{1 + x ^{6}} dx is$ (1) $tan^{- 1} x + \frac{1}{3} tan^{- 1} x^{3} + C$ (2) $tan^{- 1} x - \frac{1}{3} tan^{- 1} x^{3} + C$ (3) $tan^{- 1} x + \frac{1}{3} tan^{- 1} x^{2} + C$ (4) $tan^{- 1} x + tan^{- 1} x^{3} + C$ 98. $\int e^{sin x} \cdot (\frac{sin x + 1}{sec x}) dx is equal to$ (1) $sin x \cdot e^{sin x} + c$ (2) $cos x \cdot e^{sin x} + c$ (3) $e^{sin x} + c$ (4) $e^{sin x} (sin x + 1) + c$ 99. $\int \frac{1}{1 + e ^{x}} dx is equal to$ (1) $log_{e} (\frac{e ^{x} + 1}{e ^{x}}) + c$ (2) $log_{e} (\frac{e ^{x} - 1}{e ^{x}}) + c$ (3) $log_{e} (\frac{e ^{x}}{e ^{x} + 1}) + c$ (4) $log_{e} (\frac{e ^{x}}{e ^{x} - 1}) + c$ 100. $If \int sin^{- 1} (\frac{x}{1 + x}) dx = A (x) tan^{- 1} (x) + B (x) + C Where C is a constant of integration,then the ordered pair (A (x), B (x)) can be$ (1) $(x + 1, - x)$ (2) $(x + 1, x)$ (3) $(x - 1, - x)$ (4) $(x - 1, x)$ 101. $If \int (e^{2 x} + 2 e^{x} - e^{- x} - 1) e^{(e^{x} + e^{- x})} dx = g (x) e^{(e^{x} + e^{- x})} + c where c is a constant of integration, then g (0) is equal to :$ (1) $e^{2}$ (2) $1$ (3) $2$ (4) $e$ 102. $The value of \int_{- \frac{π}{2}}^{\frac{π}{2}} \frac{cos x}{1 + e ^{x}} dx is$ (1) $0$ (2) $1$ (3) $-2$ (4) $2$ 103. $If \frac{1}{( 3 - 5 x ) ( 2 + 3 x )} = \frac{A}{3 - 5 x} + \frac{B}{2 + 3 x}, then A : B is$ (1) $5 : 3$ (2) $2 : 3$ (3) $3 : 2$ (4) $3 : 5$ 104. $If \sum \frac{x + 5}{x ^{2} + 4 x + 5} dx = alog (x^{2} + 4 x + 5) + b tan^{- 1} (x + k) + C, then (a, b, k) equals$ (1) $(\frac{1}{2}, 3, 2)$ (2) $(\frac{1}{2}, 1, 2)$ (3) $(\frac{1}{2}, 3, 1)$ (4) $(1, 3, 2)$ 105. $If I = \int \frac{sin x + sin ^{3} x}{cos 2 x} dx = P cos x + Q log (\frac{2 cos x - 1}{2 cos x + 1}) + c_{,} (where c is a constant of integration), then the values of P and Q are respectively$ (1) $\frac{1}{2}, \frac{3}{4 2}$ (2) $\frac{1}{2}, \frac{- 3}{4 2}$ (3) $\frac{1}{2}, \frac{3}{2 2}$ (4) $\frac{1}{2}, \frac{- 3}{2 2}$ 106. $Let the equation of the curve passing through the point (0, 1) be given by y = \int x^{3} e^{x^{4}} dx . If the equation of the curve is written in the form x = f (y), then f (y) =$ (1) $log \| 4 y - 3 \|$ (2) $(log \| 4 y - 3 \|)^{1/4}$ (3) $(log (\frac{3 - 4 y}{4}))^{1/4}$ (4) $log (\frac{4 y - 3}{4})$ 107. $If I_{n} = \int_{0}^{\frac{π}{4}} tan^{n} xdx, where n is positive integer, then I_{10} + I_{8} is equal to$ (1) $9$ (2) $\frac{1}{7}$ (3) $\frac{1}{8}$ (4) $\frac{1}{9}$ 108. $If \frac{4 x ^{2} + 5 x ^{4} + 7}{( x ^{2} + 1 ) ( x ^{4} + x ^{2} + 1 )} = \frac{Ax + B}{x ^{2} + 1} + \frac{C x ^{3} + D x ^{2} + Ex + F}{x ^{4} + x ^{2} + 1}, then B + 2 (D + F + E) - C \cdot A =$ (1) $0$ (2) $3$ (3) $1$ (4) $-3$ 109. $\int e^{x} \frac{x ^{2} + 1}{( x + 1 ) ^{2}} dx is equal to$ (1) $\frac{e ^{x}}{x + 1} + C$ (2) $\frac{- e ^{x}}{x + 1} + C$ (3) $e^{x} (\frac{x - 1}{x + 1}) + C$ (4) $e^{x} (\frac{x + 1}{x - 1}) + C$ 110. $If \int \frac{x ^{49} tan ^{- 1} ( x ^{50} )}{( 1 + x ^{100} )} dx = k (tan^{- 1} (x^{50}))^{2} + c, then k =$ (1) $\frac{- 1}{100}$ (2) $\frac{1}{50}$ (3) $\frac{- 1}{50}$ (4) $\frac{1}{100}$ 111. $\sum \frac{1}{( x + \frac{2}{x} ) x ^{4} + 4 x ^{2} + 3} dx =$ (1) $\frac{1}{2} sec^{- 1} (x^{2} + 2) + c$ (2) $- cosec h^{- 1} (x^{2} + 2) + c$ (3) $\frac{1}{2} tan^{- 1} (x + \frac{2}{x}) + c$ (4) $- \frac{1}{2} cot^{- 1} (x + \frac{2}{x}) + c$ 112. $If \frac{8}{( x + 3 ) ^{2} ( x - 2 )} = \frac{Ax + B}{( x + 3 ) ^{2}} + \frac{C}{x - 2}, then 25 (B + 8 C - A) =$ (1) $25$ (2) $1$ (3) $8$ (4) $-8$ 113. $The value of \int \frac{1}{1 + cos 8 x} dx is$ (1) $\frac{tan 2 x}{8} + C$ (2) $\frac{tan 8 x}{8} + C$ (3) $\frac{tan 4 x}{4} + C$ (4) $\frac{tan 4 x}{8} + C$ 114. $Let I (x) = \int \frac{d x}{( x - 11 ) ^{\frac{11}{13}} ( x + 15 ) ^{\frac{15}{13}}} . If I (37) - I (24) = \frac{1}{4} (\frac{1}{b ^{\frac{1}{13}}} - \frac{1}{c ^{\frac{1}{13}}}), b, c \in N, then 3 (b + c) is equal to$ (1) $22$ (2) $39$ (3) $40$ (4) $26$ 115. $\int \frac{x ^{5} d x}{( x ^{2} + x + 1 ) ( x ^{6} + 1 ) ( x ^{4} - x ^{3} + x - 1 )} =$ (1) $log_{e} (\frac{x ^{6} - 1}{x ^{6} + 1}) + c$ (2) $\frac{1}{12} log_{e} (\frac{x ^{6} - 1}{x ^{6} + 1}) + c$ (3) $\frac{1}{12} log_{e} (\frac{x ^{4} + 1}{x ^{4} - 1}) + c$ (4) $log_{e} (\frac{x ^{8} + 4}{x ^{6} - 1}) + c$ 116. $If \frac{3 x + 2}{( x + 1 ) ( 2 x ^{2} + 3 )} = \frac{A}{x + 1} + \frac{Bx + C}{2 x ^{2} + 3}, then A - B + C =$ (1) $1$ (2) $2$ (3) $3$ (4) $5$ 117. $\int \frac{2 x + 2}{x ^{2} - 4 x - 5} dx is equal to$ (1) $x^{2} - 4 x - 5 + lo g (x + x^{2} - 4 x - 5) + C$ (2) $lo g (x^{2} - 4 x - 5) - x^{2} - 4 x - 5 + C$ (3) $x^{2} - 4 x - 5 + 6 lo g [(x - 2) + x^{2} - 4 x - 5] + C$ (4) $2 x^{2} - 4 x - 5 + 6 lo g [(x - 2) + x^{2} - 4 x - 5] + C$ 118. $lim n \to \infty (\frac{1 + 2 2 + 3 3 + \dots + n \cdot n}{n ^{\frac{5}{2}}}) =$ (1) $1$ (2) $\frac{5}{2}$ (3) $0$ (4) $\frac{2}{5}$ 119. $If \int \frac{1}{a ^{2} sin ^{2} x + b ^{2} cos ^{2} x} d x = \frac{1}{12} tan^{- 1} (3 tan x) + constant, then the maximum value of asin x + bcos x, is :$ (1) $40$ (2) $41$ (3) $39$ (4) $42$ 120. $If \int \frac{3 x + 1}{( x - 1 ) ( x - 2 ) ( x - 3 )} dx = A log \| x - 1 \| + B log \| x - 2 \| + C log \| x - 3 \| + c, then the values of A, B and C are respectively,$ (1) $2, - 7, - 5$ (2) $5, - 7, 5$ (3) $2, - 7, 5$ (4) $5, - 7, - 5$