Indefinite Integration PYQs Set 12 (PDF)

Set 12 of 16 · Questions 331-360 of 459

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Indefinite Integration · Set 12 (Q331-360)
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331. $The integral \int \frac{( x ^{8} - x ^{2} ) dx}{( x ^{12} + 3 x ^{6} + 1 ) tan ^{- 1} ( x ^{3} + \frac{1}{x ^{3}} )} equal to :$ (1) $log_{e} ((tan^{- 1} (x^{3} + \frac{1}{x ^{3}})))^{1/3} + C$ (2) $log_{e} ((tan^{- 1} (x^{3} + \frac{1}{x ^{3}})))^{1/2} + C$ (3) $log_{e} ((tan^{- 1} (x^{3} + \frac{1}{x ^{3}}))) + C$ (4) $log_{e} ((tan^{- 1} (x^{3} + \frac{1}{x ^{3}})))^{3} + C$ 332. $\int_{0}^{1/2} \frac{dx}{( 1 + x ^{2} ) 1 - x ^{2}} is equal to$ (1) $\frac{1}{2} tan^{- 1} \frac{2}{3}$ (2) $\frac{2}{2} tan^{- 1} (\frac{3}{2})$ (3) $\frac{2}{2} tan^{- 1} (\frac{3}{2})$ (4) $\frac{2}{2} tan^{- 1} (\frac{3}{2})$ 333. $\int x^{2} + 2 x + 5 dx is equal to$ (1) $(x + 1) x^{2} + 2 x + 5 - 2 log (x + 1 + x^{2} + 2 x + 5) + C$ (2) $\frac{1}{2} (x + 1) x^{2} + 2 x + 5 + 2 log (x + 1 + x^{2} + 2 x + 5) + C$ (3) $(x + 1) x^{2} + 2 x + 5 + 2 log (x + 1 + x^{2} + 2 x + 5) + C$ (4) $(x + 1) x^{2} + 2 x + 5 + \frac{1}{2} log (x + 1 + x^{2} + 2 x + 5) + C$ 334. $If k \int_{0}^{1} x . f (3 x) dx = \int_{0}^{3} t \cdot f (t) dt, then the value of k is$ (1) $9$ (2) $3$ (3) $1/9$ (4) $1/3$ 335. $If I_{n} = \int \frac{sin nx}{cos x} dx, then I_{n} =$ (1) $- \frac{2}{n - 1} cos (n - 1) x - I_{n - 2}$ (2) $\frac{2}{n - 1} cos (n - 1) x + I_{n - 2}$ (3) $- \frac{2}{n + 1} sin (n + 1) x - I_{n - 2}$ (4) $- \frac{2}{n + 1} cos (n - 1) x - I_{n - 2}$ 336. $If \int \frac{1 + cos 8 x}{tan 2 x - cot 2 x} dx = f (x) \cdot cos (g (x)) + c, then f (\frac{1}{4}) + g (\frac{1}{4}) =$ (1) $2$ (2) $\frac{17}{8}$ (3) $\frac{15}{8}$ (4) $\frac{33}{16}$ 337. $If [x] is the greatest integer function not greater than x, then \int_{0}^{11} [x] dx =$ (1) $55$ (2) $45$ (3) $66$ (4) $35$ 338. $If f (x) = \int \frac{1}{x ^{1/4} ( 1 + x ^{1/4} )} d x, f (0) = - 6, then f (1) is equal to:$ (1) $4 (log_{e} 2 - 2)$ (2) $2 - log_{e^{2}} 2$ (3) $log_{e} 2 + 2$ (4) $4 log_{e} 2 + 2$ 339. $If \int \frac{cos 4 x + 1}{cot x - tan x} = kcos 4 x + C, then k is equal to$ (1) $\frac{- 1}{2}$ (2) $\frac{- 1}{8}$ (3) $\frac{- 1}{3}$ (4) $\frac{- 1}{5}$ 340. $If \int e^{sec x} (sec x tan xf (x) + (sec x tan x + sec^{2} x)) dx = e^{sec x} f (x) + C then a possible choice of f (x) is:$ (1) $sec x + tan x + \frac{1}{2}$ (2) $sec x - tan x - \frac{1}{2}$ (3) $sec x + x tan x - \frac{1}{2}$ (4) $x sec x + tan x + \frac{1}{2}$ 341. $\int_{0}^{π /2} \frac{tan ^{7} x}{cot ^{7} x + tan ^{7} x} dx is equal to$ (1) $\frac{π}{4}$ (2) $\frac{π}{2}$ (3) $\frac{π}{6}$ (4) $\frac{π}{3}$ 342. $If \frac{x ^{2} + x + 1}{x ^{2} + 2 x + 1} = A + \frac{B}{x + 1} + \frac{C}{( x + 1 ) ^{2}}, then A - B is equal$ (1) $4 C$ (2) $4 C + 1$ (3) $3 C$ (4) $2 C$ 343. $If \frac{2 x + 7}{( x ^{2} + 4 ) ( x ^{2} + 9 ) ( x ^{2} + 16 )} = \frac{Ax + l}{x ^{2} + 4} + \frac{Bx + m}{x ^{2} + 9} + \frac{Cx + n}{x ^{2} + 16}, then \frac{1}{A} + \frac{1}{B} + \frac{1}{C} =$ (1) $0$ (2) $27$ (3) $\frac{105}{2}$ (4) $\frac{109}{2}$ 344. $\int_{0}^{1} \frac{dx}{e ^{x} + e ^{- x}} is equal to$ (1) $π4- tan^{- 1} (e)$ (2) $tan^{- 1} (e) -π4$ (3) $tan^{- 1} (e) +π4$ (4) $tan^{- 1} (e)$ 345. $\int \frac{25 x ^{2} + 8}{25 x ^{2} + 9} d x =$ (1) $\frac{x}{2} 25 x^{2} + 9 + \frac{11}{10} sinh^{- 1} (\frac{5 x}{3}) + C$ (2) $\frac{x}{2} 25 x^{2} + 9 - \frac{7}{10} log (\frac{5 x + 25 x ^{2} + 9}{3}) + C$ (3) $\frac{x}{2} 25 x^{2} + 9 + \frac{7}{10} sinh^{- 1} (\frac{5 x}{3}) + C$ (4) $\frac{x}{2} 25 x^{2} + 9 + \frac{11}{10} log (\frac{5 x - 25 x ^{2} + 9}{3}) + C$ 346. $\sum \frac{dx}{x + x - 1} =$ (1) $lo g_{e} ∣ x + x - 1 ∣ - \frac{1}{3} tan^{- 1} (\frac{2 x - 1 + 1}{3}) + c$ (2) $\frac{1}{3} lo g_{e} ∣ x + x - 1 ∣ - tan^{- 1} (\frac{2 x - 1 + 1}{3}) + c$ (3) $\frac{2}{3} lo g_{e} ∣ x + x - 1 ∣ - tan^{- 1} (\frac{2 x - 1 + 1}{3}) + c$ (4) $lo g_{e} ∣ x + x - 1 ∣ - \frac{2}{3} tan^{- 1} (\frac{2 x - 1 + 1}{3}) + c$ 347. $\int \frac{f ( x ) g ^{’} ( x ) - f ^{’} ( x ) g ( x )}{f ( x ) g ( x )} \times [log g (x) - log f (x)] dx is equal to$ (1) $log (\frac{g ( x )}{f ( x )}) + C$ (2) $\frac{1}{2} (log \frac{g ( x )}{f ( x )})^{2} + C$ (3) $\frac{g ( x )}{f ( x )} log (\frac{g ( x )}{f ( x )}) + C$ (4) $log (\frac{g ( x )}{f ( x )}) - \frac{g ( x )}{f ( x )} + C$ 348. $Given, \frac{3 x - 2}{( x + 1 ) ^{2} ( x + 3 )} = \frac{A}{x + 1} + \frac{B}{( x + 1 ) ^{2}} + \frac{C}{x + 3}, then 4 A + 2 B + 4 C is equal to$ (1) $5$ (2) $-5$ (3) $-3$ (4) $3$ 349. $\sum \frac{5 x ^{2} + 3}{x ^{2} ( x ^{2} - 2 )} dx =$ (1) $\frac{13}{2 2} log (\frac{2 - x}{2 + x}) + \frac{3}{2 x} + C$ (2) $\frac{13}{4 2} log (\frac{x + 2}{x - 2}) + \frac{3}{2 x} + C$ (3) $\frac{13}{4 2} log (\frac{x - 2}{x + 2}) + \frac{3}{2 x} + C$ (4) $\frac{5}{3 2} log (\frac{x + 2}{x - 2}) + \frac{3}{5} x + C$ 350. $\int \frac{( x ^{3} + 3 x ^{2} + 3 x + 1 )}{( x + 1 ) ^{5}} dx =$ (1) $- \frac{1}{( x + 1 )} + c$ (2) $15 log (x + 1) + c$ (3) $log (x + 1) + c$ (4) $tan^{- 1} x + c .$ 351. $If \int \frac{x}{x ( x + 1 )} dx = k tan^{- 1} m, then (k, m) is$ (1) $(1, x)$ (2) $(2, x)$ (3) $(2, x)$ (4) $(1, x)$ 352. $If \int \frac{cos x - sin x}{8 - sin 2 x} dx = a sin^{- 1} (\frac{sin x + cos x}{b}) + c where c is a constant of integration, then the ordered pair (a, b) is equal to :$ (1) $(1, - 3)$ (2) $(1, 3)$ (3) $(- 1, 3)$ (4) $(3, 1)$ 353. $\int \frac{1 - c o s x}{c o s x ( 1 + c o s x )} d x =$ (1) $log \| sec x + tan x \| - 2 (cosec x - cot x) + C$ (2) $log \| sec x + tan x \| - 2 (cosec x + cot x) + C$ (3) $log \| sec x + tan x \| + 2 (cosec x - cot x) + C$ (4) $log \| sec x + tan x \| + 2 (cosec x + cot x) + C$ 354. $\int_{1}^{3} \frac{4 - x}{x + 4 - x} dx =$ (1) $2$ (2) $0$ (3) $1$ (4) $3$ 355. $\int_{0}^{\frac{π}{4}} \frac{sin x + cos x}{3 + sin 2 x} dx is equal to$ (1) $\frac{1}{2} lo g 3$ (2) $log 2$ (3) $log 3$ (4) $\frac{1}{4} lo g 3$ 356. $If \int cos x \cdot cos 2 x \cdot cos 5 x d x = A sin 2 x + B sin 4 x + C sin 6 x + D sin 8 x + k (where k is the arbitrary constant of integration), then \frac{1}{B} + \frac{1}{C} =$ (1) $\frac{1}{A} - \frac{1}{D}$ (2) $\frac{1}{A} + \frac{1}{D}$ (3) $1$ (4) $0$ 357. $If \int \frac{2 sin 2 x - 3 cos x}{2 sin ^{2} x - 3 sin x + 4} dx = f (x) + c where c is the constant of integration, then f (\frac{π}{2}) - f (0) =$ (1) $2 log 2$ (2) $0$ (3) $log (\frac{3}{4})$ (4) $1$ 358. $If \frac{5 x ^{2} + 2}{x ^{3} + x} = \frac{A _{1}}{x} + \frac{A _{2} x + A _{3}}{x ^{2} + 1}, then (A_{1}, A_{2}, A_{3}) =$ (1) $(0, 2, 3)$ (2) $(3, 0, 2)$ (3) $(2, 3, 0)$ (4) $(2, 0, 3)$ 359. $Let [t] denote the greatest integer function. If \int_{0}^{2.4} [x^{2}] d x = α + β 2 + γ 3 + δ 5, then α + β + γ + δ is equal to$ 360. $Let f_{n} = \int_{0}^{\frac{π}{2}} (\sum_{k = 1}^{n} sin^{k - 1} x) (\sum_{k = 1}^{n} (2 k - 1) sin^{k - 1} x) cos x d x, n \in N . Then f_{21} - f_{20} is equal to$