Indefinite Integration PYQs Set 8 (PDF)

Set 8 of 16 · Questions 211-240 of 459

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Indefinite Integration · Set 8 (Q211-240)
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211. $For n \in N, If I_{n} = \int \frac{sin nx}{sin x} dx = \frac{2}{n - 1} sin (n - 1) x + I_{n - 2} and \int_{0}^{π} \frac{sin nx}{sin x} dx = \frac{kπ}{2} then k =$ (1) $(- 1)^{n} - 1$ (2) $1 - (- 1)^{n}$ (3) $(- 1)^{n}$ (4) $(- 1)^{n + 1}$ 212. $The integral \int \frac{( 2 x - 1 ) cos ( 2 x - 1 ) ^{2} + 5}{4 x ^{2} - 4 x + 6} dx is equal to (where c is a constant of integration)$ (1) $\frac{1}{2} sin (2 x - 1)^{2} + 5 + c$ (2) $\frac{1}{2} cos (2 x + 1)^{2} + 5 + c$ (3) $\frac{1}{2} cos (2 x - 1)^{2} + 5 + c$ (4) $\frac{1}{2} sin (2 x + 1)^{2} + 5 + c$ 213. $\int cosec (x - a) cosec xdx =$ (1) $\frac{1}{sin a} log [sin (x - a) cosec x] + C$ (2) $\frac{1}{sin a} log [sin (x - a) sin x] + C$ (3) $\frac{- 1}{sin a} log \| sin x cosec (x - a) \| + C$ (4) $\frac{- 1}{sin a} log [sin (x - a) sin x] + C$ 214. $\int \frac{dx}{sin x + cos x} is equal to$ (1) $\frac{1}{2} log (tan (\frac{x}{2} + \frac{π}{8})) + C$ (2) $\frac{1}{2} log (tan (\frac{x}{2} + \frac{π}{4})) + C$ (3) $\frac{1}{2} log (tan (\frac{x}{4} + \frac{π}{2})) + C$ (4) $\frac{1}{2} log (tan (\frac{x}{8} + \frac{π}{2})) + C$ 215. $The integral \int (\frac{x}{x sin x + cos x})^{2} dx is equal to (where C is a constant of integration)$ (1) $tan x - \frac{x sec x}{x sin x + cos x} + C$ (2) $sec x - \frac{x tan x}{x sin x + cos x} + C$ (3) $tan x + \frac{x sec x}{x sin x + cos x} + C$ (4) $sec x + \frac{x tan x}{x sin x + cos x} + C$ 216. $\int \frac{1 - x}{1 + x} dx = (where C is a constant of integration.)$ (1) $2 1 - x + cos^{- 1} x + x (1 - x)$ (2) $- 2 1 - x + cos^{- 1} x + x (1 - x)$ (3) $- 2 1 - x + cos^{- 1} x - x (1 - x)$ (4) $- 2 1 - x - cos^{- 1} x + x (1 - x)$ 217. $\int_{0}^{π /4} \frac{sin x + cos x}{3 + sin 2 x} dx =$ (1) $log 3$ (2) $\frac{1}{4} lo g 3$ (3) $2 log 3$ (4) $\frac{1}{2} log 3$ 218. $Let α \in (0, π /2) be fixed. If the integral \int \frac{tan x + tan α}{tan x - tan α} dx = A (x) cos 2 α + B (x) sin 2 α + C Where C is a constant of integration, then the functions A (x) and B (x) are respectively:$ (1) $x+α and log_{e} \| sin (x + α) \|$ (2) $x - α and log_{e} \| sin (x - α) \|$ (3) $x-α and log_{e} \| cos (x - α) \|$ (4) $x + α and log_{e} \| sin (x - α)$ 219. $If \frac{x ^{4} + 24 x ^{2} + 28}{( x ^{2} + 1 ) ^{3}} = \frac{Ax + B}{x ^{2} + 1} + \frac{Cx + D}{( x ^{2} + 1 ) ^{2}} + \frac{Ex + F}{( x ^{2} + 1 ) ^{3}} then the value of A + B + C + D + E + F =$ (1) $21$ (2) $22$ (3) $28$ (4) $29$ 220. $\int \frac{sin 4 x}{sin x} dx = (Where C is a constant of integration.)$ (1) $\frac{sin 3 x}{3} + 4 sin x + C$ (2) $\frac{1}{3} sin 3 x - \frac{2}{3} sin x + C$ (3) $\frac{2 sin 3 x}{3} + 2 sin x + C$ (4) $\frac{2}{3} sin 3 x - 2 sin x + C$ 221. $\int_{π /6}^{π /3} \frac{sin ^{3} x}{sin ^{3} x + cos ^{3} x} dx =$ (1) $\frac{π}{6}$ (2) $\frac{π}{2}$ (3) $\frac{π}{3}$ (4) $\frac{π}{12}$ 222. $Let \int \frac{2 - tan x}{3 + tan x} d x = \frac{1}{2} (αx + log_{e} \| β sin x + γ cos x \|) + C, where C is the constant of integration. Then α + \frac{γ}{β} is equal to :$ (1) $7$ (2) $4$ (3) $1$ (4) $3$ 223. $lim n \to \infty (\frac{1}{n + m} + \frac{1}{n + 2 m} + \frac{1}{n + 3 m} + \dots + \frac{1}{n + nm}) =$ (1) $\frac{log _{e} ( m )}{m}$ (2) $\frac{log _{e} ( 1 + m )}{1 + m}$ (3) $\frac{log _{e} ( 1 + m )}{m}$ (4) $\frac{log _{e} ( 1 + m )}{1 - m}$ 224. $\int \frac{( x - 1 ) e ^{x}}{( x + 1 ) ^{3}} dx =$ (1) $\frac{e ^{x}}{x + 1} + C$ (2) $\frac{e ^{x}}{( x + 1 ) ^{2}} + C$ (3) $\frac{e ^{x}}{( x + 1 ) ^{3}} + C$ (4) $\frac{x \cdot e ^{x}}{( x + 1 )} + C$ 225. $There exist θ such that a > \| sec θ \|, then \int \frac{dx}{1 + acos x} =$ (1) $\frac{1}{a ^{2} - 1} tan^{- 1} (\frac{a - 1}{a + 1} tan \frac{x}{2}) + C$ (2) $\frac{1}{a ^{2} - 1} tan^{- 1} (\frac{1 - a}{1 + a} tan \frac{x}{2}) + C$ (3) $\frac{1}{a ^{2} - 1} log (\frac{a + 1 cos \frac{x}{2} - a - 1 sin \frac{x}{2}}{a - 1 cos \frac{x}{2} + a - 1 sin \frac{x}{2}}) + C$ (4) $\frac{1}{a ^{2} - 1} log (\frac{a + 1 cos \frac{x}{2} + a - 1 sin \frac{x}{2}}{a + 1 cos \frac{x}{2} - a - 1 sin \frac{x}{2}}) + C$ 226. $\int_{- 2}^{0} (x^{3} + 3 x^{2} + 3 x + 3 + (x + 1) cos (x + 1)) dx =$ (1) $4$ (2) $0$ (3) $1$ (4) $3$ 227. $\int_{0}^{1} \frac{1 - x}{1 + x} dx is equal to$ (1) $\frac{π}{2} - 1$ (2) $\frac{π}{2} + 1$ (3) $π - 1$ (4) $\frac{3 π}{2}$ 228. $If \int \frac{x ^{3} dx}{1 + x ^{2}} = a (1 + x^{2}) 1 + x^{2} + b 1 + x^{2} + c (where c is a constant of integration), then the value of 3 ab is$ (1) $-3$ (2) $-1$ (3) $1$ (4) $3$ 229. $\int \frac{2 x - 1}{( x - 1 ) ( x + 2 ) ( x - 3 )} dx = A log \| x - 1 \| + B log \| x + 2 \| + C log \| x - 3 \| + K, then A, B, C are respectively$ (1) $\frac{- 1}{6}, \frac{1}{3}, \frac{- 1}{2}$ (2) $\frac{1}{6}, \frac{- 1}{3}, \frac{1}{3}$ (3) $\frac{1}{6}, \frac{1}{3}, \frac{1}{5}$ (4) $\frac{- 1}{6}, \frac{- 1}{3}, \frac{1}{2}$ 230. $Let f (x) = \sum \frac{( 2 - x ^{2} ) \cdot e ^{x}}{( 1 + x ) ( 1 - x ) ^{3/2}} dx . If f (0) = 0, then f (\frac{1}{2}) is equal to:$ (1) $3 e - 1$ (2) $3 e + 1$ (3) $2 e + 1$ (4) $2 e - 1$ 231. $If I = \int_{- a}^{a} (x^{4} - 2 x^{2}) dx, then I is minimum at a =$ (1) $2$ (2) $- 2$ (3) $2$ (4) $-2$ 232. $\int x^{3} sin 3 xdx =$ (1) $- \frac{x ^{3} cos 3 x}{3} - \frac{x ^{2} sin 3 x}{3} + \frac{2 x cos 3 x}{9} - \frac{2 sin 3 x}{27} + C$ (2) $- \frac{x ^{3} cos 3 x}{3} + \frac{x ^{2} sin 3 x}{3} + \frac{2 x cos 3 x}{9} - \frac{2 sin 3 x}{27} + C$ (3) $\frac{x ^{3} cos 3 x}{3} + \frac{x ^{2} sin 3 x}{3} - \frac{2 x cos 3 x}{9} - \frac{2 sin 3 x}{27} + C$ (4) $- \frac{x ^{3} cos 3 x}{3} + \frac{x ^{2} sin 3 x}{3} - \frac{2 x cos 3 x}{9} - \frac{2 sin 3 x}{27} + C$ 233. $\int \frac{s i n ( \frac{5 x}{2} )}{s i n ( \frac{x}{2} )} d x is equal to: (where c is a constant of integration)$ (1) $2 x + sin x + 2 sin 2 x + c$ (2) $x + 2 sin x + 2 sin 2 x + c$ (3) $x + 2 sin x + sin 2 x + c$ (4) $2 x + sin x + sin 2 x + c$ 234. $\int \frac{1}{x + x x} dx =$ (1) $2 log (x + 1) + C$ (2) $tan^{- 1} x + C$ (3) $\frac{1}{2} tan^{- 1} x + C$ (4) $2 tan^{- 1} x + C$ 235. $The value of \int_{0}^{4} \| x - 1 \| dx is$ (1) $\frac{5}{2}$ (2) $5$ (3) $4$ (4) $1$ 236. $\int_{0}^{\infty} \frac{6}{e ^{3 x} + 6 e ^{2 x} + 11 e ^{x} + 6} dx =$ (1) $lo g_{e} (\frac{32}{27})$ (2) $lo g_{e} (\frac{512}{81})$ (3) $lo g_{e} (\frac{256}{81})$ (4) $lo g_{e} (\frac{302}{27})$ 237. $\int \frac{e ^{x} ( 1 + sin x )}{1 + cos x} dx =$ (1) $e^{x} tan (\frac{x}{2}) + C$ (2) $e^{x} tan x + C$ (3) $c^{x} (\frac{1 + sin x}{1 - cos x}) + C$ (4) $C - e^{x} cot (\frac{x}{2})$ 238. $If \int \frac{x ^{2}}{1 - x} dx = p 1 - x (3 x^{2} + 4 x + 8) + c, where c is a constant of integration, then the value of p is$ (1) $\frac{- 2}{15}$ (2) $\frac{2}{15}$ (3) $\frac{4}{15}$ (4) $\frac{- 4}{15}$ 239. $If [.] denotes the greatest integer function, then \int_{0}^{1000} e^{x - [x]} dx =$ (1) $\frac{e ^{1000} - 1}{1000}$ (2) $1000 (e - 1)$ (3) $\frac{e ^{1000} - 1}{e - 1}$ (4) $\frac{e - 1}{1000}$ 240. $\sum \frac{dx}{x ^{3} + 3 x ^{2} + 2 x} =$ (1) $log \| x \| + log (\frac{x + 2}{x + 1}) + c$ (2) $log \| x \| - log \| x + 1 \| + log \| x + 2 \| + c$ (3) $\frac{1}{2} [log \| x \| + log \| x + 1 \| + log \| x + 2 \|] + c$ (4) $\frac{1}{2} log (\frac{( x ^{2} + 2 x )}{( x + 1 ) ^{2}}) + c$