Indefinite Integration PYQs Set 13 (PDF)

Set 13 of 16 · Questions 361-390 of 459

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Indefinite Integration · Set 13 (Q361-390)
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361. $The sum of the roots of the equation x + 1 - 2 l o g_{2} (3 + 2^{x}) + 2 l o g_{4} (10 - 2^{- x}) = 0, is :$ (1) $l o g_{2} 14$ (2) $l o g_{2} 11$ (3) $log2 12$ (4) $l o g_{2} 13$ 362. $The integral \int cos (lo g_{e} x) d x is equal to : (where C is a constant of integration$ (1) $x [cos (lo g_{e} x) - sin (lo g_{e} x)] + C$ (2) $\frac{x}{2} [sin (lo g_{e} x) - cos (lo g_{e} x)] + C$ (3) $x [cos (l o g_{e} x) + s in (l o g_{e} x)] + C$ (4) $\frac{x}{2} [cos (lo g_{e} x) + sin (lo g_{e} x)] + C$ 363. $Let I_{n} = \int tan^{n} x d x, (n > 1). If I_{4} + I_{6} = a t a n^{5} x + b x^{5} + C, where C is a constant of integration, then the ordered pair (a, b) is equal to$ (1) $(- \frac{1}{5}, 1)$ (2) $(\frac{1}{5}, 0)$ (3) $(\frac{1}{5}, - 1)$ (4) $(- \frac{1}{5}, 0)$ 364. $The integral \int \frac{s e c ^{2} x}{( s e c x + t a n x ) ^{9/2}} d x equals (for some arbitrary constant K)$ (1) $\frac{-1}{( sec x + t an x ) ^{11/2}} + K$ (2) $\frac{1}{( sec x + t an x ) ^{11/2}} + K$ (3) $\frac{-1}{( sec x + t an x ) ^{11/2}} + K$ (4) $\frac{1}{( sec x + t an x ) ^{11/2}} + K$ 365. $\int_{- π /4}^{π /4} \frac{dx}{1 + cos 2 x} is equal to$ (1) $1$ (2) $0$ (3) $2$ (4) $4$ 366. $If I_{n} = \int_{π /2}^{\infty} e^{- x} cos^{n} x d x, then \frac{I _{2018}}{I _{2016}} =$ (1) $\frac{2018 \times 2019}{( 2017 ) ^{2} + 1}$ (2) $\frac{2018 \times 2017}{( 2018 ) ^{2} + 1}$ (3) $\frac{( 2018 ) ( 2016 )}{( 2017 ) ^{2} + 1}$ (4) $\frac{( 2018 ) ( 2017 )}{( 2019 ) ^{2} + 1}$ 367. $\int_{0}^{π /4} \frac{c o s ^{2} x}{c o s ^{2} x + 4 s i n ^{2} x} d x =$ (1) $\frac{π}{4} + \frac{2}{3} tan^{- 1} 2$ (2) $- \frac{π}{3} - \frac{2}{3} tan^{- 1} 3$ (3) $- \frac{π}{12} + \frac{2}{3} tan^{- 1} 2$ (4) $\frac{π}{6} - \frac{2}{3} tan^{- 1} 4$ 368. $The value of \int_{- 1/2}^{1/2} cos^{- 1} x d x is$ (1) $\frac{π}{2}$ (2) $1$ (3) $\frac{π ^{2}}{2}$ (4) $π$ 369. $\int_{0}^{π} \frac{x d x}{4 c o s ^{2} x + 9 s i n ^{2} x} =$ (1) $\frac{π ^{2}}{12}$ (2) $\frac{π ^{2}}{4}$ (3) $\frac{π ^{2}}{6}$ (4) $\frac{π ^{2}}{3}$ 370. $\int_{2}^{5} (x + 2 x - 1 + x - 2 x - 1) d x =$ (1) $16/3$ (2) $32/3$ (3) $28/3$ (4) $4/3$ 371. $\int_{0}^{4} ∣ x - 2∣ - x d x =$ (1) $2$ (2) $3$ (3) $6$ (4) $12$ 372. $\int_{3}^{6} \frac{x}{9 - x + x} d x =$ (1) $\frac{1}{2}$ (2) $\frac{3}{2}$ (3) $2$ (4) $1$ 373. $\int_{0}^{π} (cos^{2} (\frac{3 π}{8} - \frac{x}{4}) - cos^{2} (\frac{11 π}{8} + \frac{x}{4})) d x =$ (1) $1/ 2$ (2) $22$ (3) $2$ (4) $2$ 374. $\int_{0}^{π} \frac{x}{s i n x} (3 cos^{2} x + 2 sin x + sin^{3} x - 3) d x =$ (1) $\frac{π ( 5 π - 12 )}{4}$ (2) $\frac{π}{2}$ (3) $\frac{π}{2} (5 π - 6)$ (4) $\frac{π ( 5 π - 12 )}{6}$ 375. $lim_{n \to \infty} \prod_{r = 1}^{n} (1 + \frac{r ^{2}}{n ^{2}})^{2 r / n^{2}} =$ (1) $log (\frac{4}{e})$ (2) $log (\frac{2}{e})$ (3) $\frac{2}{e}$ (4) $\frac{4}{e}$ 376. $The parametric form of a curve is x = \frac{t ^{3}}{t ^{2} - 1}, y = \frac{t}{t ^{2} - 1}, then \int \frac{d x}{x - 3 y} =$ (1) $\frac{1}{2} log (t^{2} - 1) + C$ (2) $2 log (t (t^{2} - 1)) + C$ (3) $\frac{1}{4} log (\frac{t}{t ^{2} - 3}) + C$ (4) $\frac{5}{2} log (t + \frac{1}{t ^{2}}) + C$ 377. $Given that \frac{d}{d x} (\int_{0}^{φ (x)} f (t) d t) = f (φ (x)) φ^{'} (x) . If \int_{0}^{x^{3}} f (t) d t = x^{2} sin 2 π x, then the value of f (8) is$ (1) $\frac{2 π}{3}$ (2) $\frac{4 π}{3}$ (3) $\frac{π}{3}$ (4) $\frac{π}{12}$ 378. $If A (x) = 1 x + 1 x^{2} + 1 2 2 x + 1 2 x^{2} + 1 3 3 x + 1 3 x^{2} + 1, then \int_{0}^{1} A (x) d x equals$ (1) $0$ (2) $1$ (3) $2$ (4) $4$ 379. $\int_{0}^{π /2} \frac{s i n ( \frac{π}{4} + x ) + s i n ( \frac{3 π}{4} + x )}{c o s x + s i n x} d x =$ (1) $\frac{π}{2}$ (2) $\frac{π}{2 2}$ (3) $\frac{π}{3 2}$ (4) $\frac{π}{4 2}$ 380. $If I_{n} = \int_{0}^{π /4} tan^{n} x d x, where n is a positive integer, then I_{10} + I_{8} is$ (1) $\frac{1}{9}$ (2) $\frac{1}{8}$ (3) $\frac{1}{7}$ (4) $9$ 381. $If f (x) = \int_{1}^{x} \frac{1}{2 + t ^{x}} dt, then$ (1) $\frac{1}{18} < f (2) < \frac{1}{3}$ (2) $f (2) < \frac{1}{2} (or) f (2) > 2$ (3) $f (2) < \frac{1}{3}$ (4) $f (2) > \frac{1}{3}$ 382. $\int_{0}^{α /3} \frac{f ( x )}{f ( x ) + f ( \frac{α - 3 x}{3} )} d x =$ (1) $\frac{2 α}{3}$ (2) $\frac{α}{2}$ (3) $\frac{α}{3}$ (4) $\frac{α}{6}$ 383. $\int_{0}^{1/2} \frac{x s i n ^{- 1} x}{1 - x ^{2}} d x =$ (1) $(\frac{1}{2} + \frac{3}{12} π)$ (2) $(\frac{1}{2} - \frac{3}{12} π)$ (3) $(- \frac{1}{2} + \frac{3}{12} π)$ (4) $(- \frac{1}{2} - \frac{3}{12} π)$ 384. $\int_{0}^{1} \frac{l o g _{e} ( 1 + x )}{1 + x ^{2}} d x =$ (1) $\frac{π}{4} log_{e} 2$ (2) $\frac{π}{6} log_{e} 6$ (3) $\frac{π}{2} log_{e} 8$ (4) $\frac{π}{8} log_{e} 2$ 385. $tan^{- 1} (\int_{- π /2}^{π /2} \frac{c o s x}{1 + e ^{x}} d x) =$ (1) $\frac{π}{4}$ (2) $\frac{π}{3}$ (3) $\frac{π}{6}$ (4) $\frac{π}{2}$ 386. $\int_{1/25}^{3} \frac{e ^{3/ x}}{x ^{2}} d x =$ (1) $- \frac{1}{3} (e^{75} - e)$ (2) $\frac{1}{3} (e^{50} - e^{25})$ (3) $- \frac{1}{3} (e^{50} - e)$ (4) $\frac{1}{3} (e^{75} - e)$ 387. $Let f (x) = \int \frac{2 x}{( x ^{2} + 1 ) ( x ^{2} + 3 )} d x . If f (3) = \frac{1}{2} (lo g_{e} 5 - lo g_{e} 6), then f (4) is equal to$ (1) $\frac{1}{2} (lo g_{e} 17 - lo g_{e} 19)$ (2) $log_{e} 17 - log_{e} 19$ (3) $\frac{1}{2} (lo g_{e} 19 - lo g_{e} 18)$ (4) $log_{e} 19 - log_{e} 20$ 388. $If \int \frac{2 e ^{x} + 3 e ^{- x}}{4 e ^{x} + 7 e ^{- x}} d x = \frac{1}{4} (ux + v lo g_{e} (4 e^{x} + 7 e^{- x})) + C, where C is a constant of integration, then u+v is equal to .$ 389. $If \int sec 2 x - 1 d x = α lo g_{e} cos 2 x + β + cos 2 x (1 + cos 2 x) + constant, then β - α is equal to ______$ 390. $If \int \frac{1}{5 ( x - 1 ) ^{4} ( x + 3 ) ^{6}} d x = A (\frac{α x - 1}{β x + 3})^{B} + C, where C is the constant of integration, then the value of α + β + 20 A B is_______$