Indefinite Integration PYQs Set 11 (PDF)

Set 11 of 16 · Questions 301-330 of 459

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Indefinite Integration · Set 11 (Q301-330)
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301. $Given that \int \frac{1}{x ^{2} + a ^{2}} dx = \frac{1}{a} tan^{- 1} (\frac{x}{a}) + c . If \int \frac{1}{x ^{4} + 3 x ^{2} + 1} dx = a \cdot tan^{- 1} (\frac{b ( x ^{2} - 1 )}{x}) + c tan^{- 1} (\frac{d ( x ^{2} + 1 )}{x}) + k where k is a constant of integration, then 5 (c + d + ab) =$ (1) $3$ (2) $5$ (3) $8$ (4) $10$ 302. $If I_{1} = \int_{0}^{π /2} x sin xdx and I_{2} = \int_{0}^{π /2} x cos xdx, then which one of the following is true?$ (1) $I_{1} + I_{2} = \frac{π}{2}$ (2) $I_{1} = \frac{π}{2} I_{2}$ (3) $I_{1} + I_{2} = 0$ (4) $I_{1} = I_{2} .$ 303. $\int e^{x} - 4 dx equals$ (1) $tan^{- 1} (\frac{e ^{x} - 4}{2}) + e^{x} - 4 + C$ (2) $2 e^{x} - 4 - 4 tan^{- 1} (\frac{e ^{x} - 4}{2}) + C$ (3) $2 e^{x} - 4 - 4 cot^{- 1} (\frac{e ^{x} - 4}{2}) + C$ (4) $e^{x} - 4 - 4 tan^{- 1} (e^{x} - 4) + C$ 304. $Let I (x) = \int \frac{6}{sin ^{2} x ( 1 - cot x ) ^{2}} dx . If I (0) = 3, then I (\frac{π}{12}) is equal to$ (1) $23$ (2) $3$ (3) $33$ (4) $63$ 305. $\int \frac{dx}{cos ( x + 4 ) cos ( x + 2 )} is equal to$ (1) $\frac{1}{sin 2} log (cos (x + 4)^{2}) + C$ (2) $\frac{1}{2} log (\frac{sec ( x + 2 )}{sec ( x + 4 )}) + C$ (3) $\frac{1}{sin 2} log (\frac{sec ( x + 4 )}{sec ( x + 2 )}) + C$ (4) $log (\frac{sec ( x + 4 )}{sec ( x + 2 )}) + C$ 306. $\int \frac{1 + t a n x t a n ( x + α )}{t a n x t a n ( x + α )} d x =$ (1) $tan α (log (sec (x + a)) + log sec x + C)$ (2) $cot α (log \| sin x \| - log \| sin (x + α) \|) + C$ (3) $tan α (log (\frac{cos x}{sin ( x + α )})) + C$ (4) $cot α (log \frac{sin ( x + a )}{cos ( x + α )}) + C$ 307. $\int \frac{x + 1}{x ( 1 + x e ^{x} )} dx is equal to$ (1) $log (\frac{1 + x e ^{x}}{x e ^{x}}) + C$ (2) $log (\frac{x e ^{x}}{1 + x e ^{x}}) + C$ (3) $log (x e^{x} (1 + x e^{x})) + C$ (4) $log (1 + x e^{x}) + C$ 308. $\int \frac{( x + 3 ) e ^{x}}{( x + 4 ) ^{2}} dx is equal to$ (1) $\frac{e ^{x}}{( x + 4 )} + C$ (2) $\frac{e ^{x}}{( x + 4 ) ^{2}} + C$ (3) $\frac{e ^{x}}{( x + 3 )} + C$ (4) $\frac{1}{( x + 4 ) ^{2}} + C$ 309. $\sum \frac{cos 2 x \cdot sin 4 x}{cos ^{4} x ( 1 + cos ^{2} 2 x )} dx =$ (1) $log (\frac{1 + cos 2 x}{1 + cos ^{2} 2 x}) + sec^{2} x + c$ (2) $log \frac{( 1 + cos 2 x ) ^{2}}{( 1 + cos ^{2} x )} + sec x + c$ (3) $log \frac{( 1 + cos 2 x ) ^{2}}{( 1 + cos ^{2} 2 x )} + sec^{2} x + c$ (4) $log \frac{1 + cos ^{2} 2 x}{( 1 + cos 2 x ) ^{2}} + sec x + c$ 310. $For n \geq 2, If I_{n} = \int sec^{n} xdx, then I_{4} - \frac{2}{3} I_{2} =$ (1) $sec^{2} xtan x + c$ (2) $13 sec^{2} xtan x + c$ (3) $23 sec^{2} xtan x + c$ (4) $\frac{1}{2} log \| sec x + tan x \| + c$ 311. $If \int \frac{x}{( a + x ) ^{5}} dx = \frac{1}{k ( a + x ) ^{4}} (f (x)) + c then \frac{f ( - a )}{ak} =$ (1) $\frac{1}{3}$ (2) $\frac{1}{2}$ (3) $\frac{5}{6}$ (4) $\frac{1}{4}$ 312. $If \int_{- a}^{a} f (x) dx = \int_{0}^{a} f (x) dx + \int_{0}^{a} g (x) dx then g (x) =$ (1) $- f (x)$ (2) $f (x)$ (3) $f (- x)$ (4) $f (x) + f (- x)$ 313. $If f (x) = f (π + e - x) and \int_{e}^{π} f (x) dx = \frac{2}{e + π}, then \int_{e}^{π} xf (x) dx is equal to$ (1) $π - e$ (2) $\frac{π + e}{2}$ (3) $1$ (4) $π - e 2$ 314. $The value of \int \frac{e ^{6 log x} - e ^{5 log x}}{e ^{4 log x} - e ^{3 log x}} dx is equal to$ (1) $0$ (2) $\frac{x ^{3}}{3}$ (3) $\frac{3}{x ^{3}}$ (4) $\frac{1}{x}$ 315. $If \int \frac{cos xdx}{sin ^{3} x ( 1 + sin ^{6} x ) ^{2/3}} = f (x) (1 + sin^{6} x)^{1/ λ} + c where c is a constant of integration, then λf (\frac{π}{3}) is equal to:$ (1) $4$ (2) $- 2$ (3) $8$ (4) $- 4$ 316. $\int \frac{sin x cos x}{1 - sin ^{4} x} dx =$ (1) $tan^{- 1} (sin^{2} x) + C$ (2) $tan^{- 1} (2 sin x) + C$ (3) $\frac{1}{2} sin^{- 1} (sin^{2} x) + C$ (4) $\frac{1}{2} cos^{- 1} (sin^{2} x) + C$ 317. $If \int \frac{x ^{3}}{1 + x ^{2}} dx = A (1 + x^{2})^{\frac{3}{2}} + B (1 + x^{2})^{\frac{1}{2}} + C, then A + B =$ (1) $\frac{2}{3}$ (2) $- \frac{2}{3}$ (3) $\frac{1}{3}$ (4) $- \frac{1}{3}$ 318. $\sum (log (log x) + \frac{1}{( log x ) ^{2}}) dx =$ (1) $x [log (log x) + log x] + c$ (2) $\frac{x}{log ( log x )} + c$ (3) $xlog (log x) + c$ (4) $x (log (log x) - \frac{1}{log x}) + c$ 319. $If a \in Z^{+}, [x] is greatest integer not more than x and \int_{0}^{a} 2^{[x]} dx = 127, then a =$ (1) $6$ (2) $7$ (3) $8$ (4) $9$ 320. $The value of \int_{- π /4}^{π /4} sin^{103} x \cdot cos^{101} xdx is$ (1) $(\frac{π}{4})^{103}$ (2) $(\frac{π}{4})^{101}$ (3) $2$ (4) $0$ 321. $If \frac{3}{( x - 1 ) ( x ^{2} + x + 1 )} = \frac{1}{x - 1} - \frac{x + 2}{x ^{2} + x + 1} = f_{1} (x) - f_{2} (x) \frac{x + 1}{( x - 1 ) ^{2} ( x ^{2} + x + 1 )} = A f_{1} (x) + (B + \frac{D}{x - 1}) f_{2} (x) + \frac{C}{( x - 1 ) ^{2}} A + B + C + D =$ (1) $1$ (2) $\frac{- 1}{3}$ (3) $0$ (4) $\frac{1}{3}$ 322. $If \int f (x) cos xdx = \frac{1}{2} (f (x))^{2} + C and f (0) = 0, then f^{’} (0) = ?$ (1) $1$ (2) $-1$ (3) $0$ (4) $2$ 323. $\sum \frac{sin ^{3} ( x ) + cos ^{3} ( x )}{sin ^{2} ( x ) \cdot cos ^{2} ( x )} dx =$ (1) $sec (x) - cosec (x) + c$ (2) $tan (x) + cot (x) + c$ (3) $cosec (x) - cot (x) + c$ (4) $tan (x) - cot (x) + c$ 324. $If the partial fraction decomposition of$ (1) $0$ (2) $2$ (3) $3$ (4) $4$ 325. $\int_{0}^{π /2} \frac{16 x sin x cos x}{sin ^{4} x + cos ^{4} x} dx is equal to$ (1) $\frac{π ^{2}}{4}$ (2) $\frac{π ^{2}}{2}$ (3) $π^{2}$ (4) $2 π^{2}$ 326. $\int \frac{1}{( s i n x + c o s x + 2 s i n 2 x ) ^{2}} d x =$ (1) $\frac{- ( 1 + 3 tan x )}{( 3 + tan ^{2} x ) ^{3}} + C$ (2) $\frac{- ( 1 + 3 tan x )}{3 ( 1 + tan x ) ^{3}} + C$ (3) $\frac{- ( 1 + tan x )}{3 ( 1 + 3 tan x ) ^{2}} + C$ (4) $\frac{( 1 + 3 tan x )}{( 3 + tan ^{2} x ) ^{3}} + C$ 327. $If \int log (a^{2} + x^{2}) dx = h (x) + C, then h (x) is equal to$ (1) $xlog (a^{2} + x^{2}) + 2 tan^{- 1} (\frac{x}{a})$ (2) $x^{2} log (a^{2} + x^{2}) + x + a tan^{- 1} (\frac{x}{a})$ (3) $xlog (a^{2} + x^{2}) - 2 x + 2 a tan^{- 1} (\frac{x}{a})$ (4) $x^{2} log (a^{2} + x^{2}) + 2 x - a^{2} tan^{- 1} (\frac{x}{a})$ 328. $\int_{2}^{8} \frac{5 ^{10 - x}}{5 ^{x} + 5 ^{10 - x}} dx =$ (1) $4$ (2) $5$ (3) $3$ (4) $6$ 329. $f f (x) = \int \frac{5 x ^{8} + 7 x ^{6}}{( x ^{2} + 1 + 2 x ^{7} ) ^{2}} dx, (x \geq 0), f (0) = 0 v and f (1) = \frac{1}{K}, then the value of K is$ 330. $If \int (sin x)^{- \frac{11}{2}} (cos x)^{- \frac{5}{2}} d x = - \frac{p _{1}}{q _{1}} (cot x)^{\frac{9}{2}} - \frac{p _{2}}{q _{2}} (cot x)^{\frac{5}{2}} - \frac{p _{3}}{q _{3}} (cot x)^{\frac{1}{2}} + \frac{p _{4}}{q _{4}} (cot x)^{- \frac{3}{2}} + C, where p_{i} and q_{i} are positive integers with g cd (p_{i}, q_{i}) = 1 for i = 1, 2, 3, 4 and C is the constant of integration, then \frac{15 p _{1} p _{2} p _{3} p _{4}}{q _{1} q _{2} q _{3} q _{4}} is equal to$