Indefinite Integration PYQs Set 9 (PDF)

Set 9 of 16 · Questions 241-270 of 459

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Indefinite Integration · Set 9 (Q241-270)
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241. $If f (x) = \int x^{2} cos^{2} x (2 x tan^{2} x - 2 x - 6 tan x) dx and f (0) = π then f (x) =$ (1) $x^{2} sin x + π$ (2) $cos x + π - 1$ (3) $- x^{3} sin 2 x + π$ (4) $x^{3} cos 2 x + πcos x$ 242. $If \int \frac{dx}{x ^{3} ( 1 + x ^{6} ) ^{2/3}} = xf (x) (1 + x^{6})^{\frac{1}{3}} + C Where C is a constant of integration, then the function f (x) is equal to:$ (1) $\frac{3}{x ^{2}}$ (2) $- \frac{1}{6 x ^{3}}$ (3) $- \frac{1}{2 x ^{2}}$ (4) $- \frac{1}{2 x ^{3}}$ 243. $If \int \frac{( x - 1 ) ^{2}}{( x ^{2} + 1 ) ^{2}} dx = tan^{- 1} (x) + g (x) + k, then g (x) is equal to$ (1) $tan^{- 1} (\frac{x}{2})$ (2) $\frac{1}{x ^{2} + 1}$ (3) $\frac{1}{2 ( x ^{2} + 1 )}$ (4) $\frac{2}{x ^{2} + 1}$ 244. $If \int x^{5} e^{- 4 x^{3}} dx = \frac{1}{48} e^{- 4 x^{3}} f (x) + C Where c is constant of integration, then f (x) is equal to:$ (1) $- 2 x^{3} - 1$ (2) $- 4 x^{3} - 1$ (3) $- 2 x^{3} + 1$ (4) $4 x^{3} + 1$ 245. $\int e^{x} (\frac{sin x + cos x}{1 - sin ^{2} x}) dx is$ (1) $(e^{x} \cdot cosec x) + c$ (2) $e^{x} cot x + c$ (3) $(e^{x} \cdot sec x) + c$ (4) $e^{x} tan x + c$ 246. $\sum e^{x} (\frac{sec ^{2} x + tan x - cot x}{sin x}) dx =$ (1) $e^{x} (cosec x + cot x) + c$ (2) $e^{x} (\frac{sec ^{2} x + tan x}{sin x}) + c$ (3) $e^{x} (\frac{1 + tan ^{2} x}{sin x}) + c$ (4) $e^{x} (cosec x + sec x) + c$ 247. $\int \frac{x}{( x ^{2} + 2 x + 2 ) ^{2}} dx is equal to$ (1) $\frac{x ^{2} + 2}{x ^{2} + 2 x + 2} - \frac{1}{2} tan^{- 1} (x + 1) + C$ (2) $\frac{x ^{2} + 2}{2 ( x ^{2} + 2 x + 2 )} - \frac{1}{2} tan^{- 1} (x - 1) + C$ (3) $- \frac{x ^{2} - 2}{4 ( x ^{2} + 2 x + 2 )} - \frac{1}{2} tan^{- 1} (x + 1) + C$ (4) $\frac{2 ( x - 1 )}{x ^{2} + 2 x + 2} + \frac{1}{2} tan^{- 1} (x + 1) + C$ 248. $\int \frac{sin x + sin ^{3} x}{cos 2 x} dx = A cos x + B log f (x) + c (where c is a constant of integration). Then values of A, B and f (x) are$ (1) $A = \frac{1}{2}, B = \frac{- 3}{4 2}, f (x) = \frac{2 cos x - 1}{2 cos x + 1}$ (2) $A = - \frac{1}{2}, B = \frac{- 3}{4 2}, f (x) = \frac{2 cos x + 1}{2 cos x - 1}$ (3) $A = \frac{1}{2}, B = \frac{- 3}{4 2}, f (x) = \frac{2 cos x + 1}{2 cos x - 1}$ (4) $A = \frac{3}{2}, B = \frac{1}{2}, f (x) = \frac{2 cos x - 1}{2 cos x + 1}$ 249. $The integral \int \frac{e ^{3 log_{e} 2 x} + 5 e ^{2 log_{e} 2 x}}{e ^{4 log_{e} x} + 5 e ^{3 log_{e} x} - 7 e ^{2 log_{e} x}} dx, x > 0, is equal to: (where " c" is a constant of integration)$ (1) $log_{e} (x^{2} + 5 x - 7) + c$ (2) $\frac{1}{4} log_{e} (x^{2} + 5 x - 7) + c$ (3) $4 log_{e} (x^{2} + 5 x - 7) + c$ (4) $log_{e} x^{2} + 5 x - 7 + c$ 250. $\int \frac{x \cdot l o g x}{( x ^{2} - 1 ) ^{3}} d x =$ (1) $sec^{- 1} x + \frac{log x}{x ^{2} - 1} + C$ (2) $sec^{- 1} x - \frac{log x}{x ^{2} - 1} + C$ (3) $\frac{log x}{x ^{2} - 1} - sec^{- 1} x + C$ (4) $\frac{- log x}{x ^{2} - 1} - sec^{- 1} x + C$ 251. $\int \frac{x ^{2} - 2}{x ^{3} x ^{2} - 1} d x =$ (1) $\frac{- x ^{2} - 1}{x}$ (2) $-x2-2x$ (3) $\frac{- x}{x ^{2} - 1}$ (4) $\frac{- x ^{2} - 1}{x ^{2}}$ 252. $\sum \frac{x ^{4} + x ^{- 4} + 2}{x ^{3}} dx =$ (1) $log \| x \| - \frac{1}{4 x ^{4}} + C$ (2) $log \| x \| + \frac{1}{4 x ^{4}} + C$ (3) $log \| x \| - \frac{4}{x ^{4}} + C$ (4) $log \| x \| + \frac{4}{x ^{4}} + C$ 253. $lim n \to \infty \frac{1}{n} (1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \dots + \frac{1}{n}) =$ (1) $2$ (2) $2$ (3) $2 - 1$ (4) $22$ 254. $\int \frac{x}{x + 1 + x - 1} d x = A (x) (x + 1)^{\frac{3}{2}} + B (x) (x - 1)^{\frac{3}{2}} + C, then A (x) + B (x) =$ (1) $\frac{4}{15}$ (2) $- \frac{4}{15}$ (3) $\frac{2 x}{5}$ (4) $- \frac{2 x}{5}$ 255. $If \frac{x ^{2} + 3}{( x ^{2} + 1 ) ( x ^{2} + 2 )} = \frac{Ax + B}{x ^{2} + 1} + \frac{Cx + D}{x ^{2} + 2} then A + B + C + D =$ (1) $3$ (2) $2$ (3) $0$ (4) $1$ 256. $If \frac{3 π}{4} < x < \frac{7 π}{4} then \int (2^{x} - 1 + sin 2 x + \frac{1}{x ^{2}} - \frac{1}{x}) dx =$ (1) $\frac{2 ^{x}}{log 2} - sin x + cos x - \frac{1}{x} - log x + c$ (2) $2^{x} log 2 + sin x - cos x - \frac{1}{x} + \frac{1}{x ^{2}} + c$ (3) $\frac{2 ^{x}}{log 2} + sin x - cos x - \frac{1}{x} - log x + c$ (4) $2^{x} log 2 - sin x + cos x - \frac{1}{x} + \frac{1}{x ^{2}} + c$ 257. $If \frac{x ^{2} - 3 x + 2}{( x - 4 ) ( x - 3 ) ^{2}} = \frac{A}{x - 4} + \frac{B}{x - 3} + \frac{C}{( x - 3 ) ^{2}} then A + B + C =$ (1) $1$ (2) $0$ (3) $-1$ (4) $5$ 258. $By the definition of the definite integral, the value of lim n \to \infty (\frac{1}{n ^{2} - 1} + \frac{1}{n ^{2} - 2 ^{2}} + \dots + \frac{1}{n ^{2} - ( n - 1 ) ^{2}}) is equal to$ (1) $π$ (2) $\frac{π}{2}$ (3) $\frac{π}{4}$ (4) $\frac{π}{6}$ 259. $If \int f (x) dx = g (x), then \int f (x) g (x) dx =$ (1) $\frac{1}{2} (g^{’} (x))^{2}$ (2) $f^{’} (x) g (x)$ (3) $\frac{1}{2} f^{2} (x)$ (4) $\frac{1}{2} g^{2} (x)$ 260. $\int_{- 1/2}^{1/2} ([x] + log (\frac{1 + x}{1 - x})) dx is equal to$ (1) $2 log (\frac{1}{2})$ (2) $0$ (3) $\frac{- 1}{2}$ (4) $1$ 261. $\int \frac{sin ^{2} x}{1 + cos x} dx is equal to$ (1) $x - sin x + C$ (2) $cos x + C$ (3) $x + sin x + C$ (4) $sin x + C$ 262. $\int \frac{cos 2 x - cos 2 α}{cos x - cos α} dx is equal to$ (1) $2 (sin x - x cos α) + c$ (2) $2 (sin x - 2 x cos α) + c$ (3) $2 (sin x + x cos α) + c$ (4) $2 (sin x + 2 x cos x) + c$ 263. $\int \frac{x}{1 - 2 x ^{4}} dx = (where C is a constant of integration)$ (1) $\frac{1}{2} sin^{- 1} (2 x) + C$ (2) $\frac{1}{2 2} sin^{- 1} (2 x^{2}) + C$ (3) $\frac{1}{2 2} sin^{- 1} (22 x^{2}) + C$ (4) $\frac{1}{2} sin^{- 1} (2 x) + C$ 264. $\int (tan x + cot x) d x =$ (1) $2 sin^{- 1} (sin x + cos x) + c$ (2) $2 cos (sin x cos x) + c$ (3) $2 cos^{- 1} (sin x - cos x) + c$ (4) $2 sin^{- 1} (sin x - cos x) + c$ 265. $If lim_{n \to \infty} ((1 + \frac{1}{n ^{2}}) (1 + \frac{2 ^{2}}{n ^{2}}) \dots (1 + \frac{n ^{2}}{n ^{2}}))^{\frac{1}{n}} = k then log k =$ (1) $log 4 + \frac{π}{2} - 1$ (2) $log 2 + \frac{π}{2} + 1$ (3) $log 2 + \frac{π}{2} - 2$ (4) $log 2 + \frac{π}{2} - 1$ 266. $For x \in (- \frac{π}{2}, \frac{π}{2}), if y (x) = \int \frac{cosec x + sin x}{cosec x sec x + tan x sin ^{2} x} dx and lim_{x \to (\frac{π}{2})^{-}} y (x) = 0 then y (\frac{π}{4}) is equal to$ (1) $tan^{- 1} (\frac{1}{2})$ (2) $\frac{1}{2} tan^{- 1} (\frac{1}{2})$ (3) $- \frac{1}{2} tan^{- 1} (\frac{1}{2})$ (4) $\frac{1}{2} tan^{- 1} (- \frac{1}{2})$ 267. $The value of \int_{- 1}^{2} \frac{\| x \|}{x} dx is$ (1) $0$ (2) $1$ (3) $2$ (4) $3$ 268. $\int (\frac{8 ^{1 + x} + 4 ^{1 + x}}{2 ^{2 x}}) d x =$ (1) $\frac{2 ^{x}}{log 2} + 4 x + C$ (2) $8 \cdot \frac{2 ^{x}}{log 2} - 4 x + C$ (3) $8 \cdot \frac{2 ^{x}}{log 2} + 4 x + C$ (4) $\frac{2 ^{x}}{log 2} - 4 x + C$ 269. $If \int \frac{dx}{x ^{2022} ( 1 + x ^{2022} ) ^{1/2022}} = \frac{- ( 1 + x ^{m} ) ^{n / m}}{n x ^{n}} + C then m - n =$ (1) $1$ (2) $2$ (3) $3$ (4) $0$ 270. $If \int f (x) dx = ψ (x), then \int x^{5} f (x^{3}) dx =$ (1) $\frac{1}{3} (x^{3} ψ (x^{3})) - \int x^{2} ψ (x^{3}) dx$ (2) $\frac{1}{3} (x^{3} ψ (x^{3})) + \int x^{2} ψ (x^{3}) dx$ (3) $- \frac{1}{3} (x^{3} ψ (x^{3})) - \int x^{3} ψ (x^{3}) dx$ (4) $- \frac{1}{3} (x^{3} ψ (x^{3})) + \int x^{3} ψ (x^{3}) dx$