Indefinite Integration PYQs Set 10 (PDF)

Set 10 of 16 · Questions 271-300 of 459

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Indefinite Integration · Set 10 (Q271-300)
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271. $If \int_{3}^{b} \frac{x - 1}{2 x - x ^{2}} dx = \frac{1}{2}, then (b - 1)^{2} =$ (1) $2$ (2) $2$ (3) $1 + \frac{3}{e}$ (4) $\frac{3}{e} - 1$ 272. $\int (x + 2) x + 3 dx is equal to$ (1) $\frac{2}{15} x + 3 (3 x^{2} - 13 x + 12) + C$ (2) $\frac{2}{15} x + 3 (3 x^{2} + 13 x + 12) + C$ (3) $\frac{2}{5} x + 3 (3 x^{2} - 12 x + 13) + C$ (4) $\frac{2}{5} x + 3 (3 x^{2} + 12 x + 13) + C$ 273. $If f (x) = \int \frac{5 x ^{8} + 7 x ^{6}}{( x ^{2} + 1 + 2 x ^{7} ) ^{2}} dx, (x \geq 0) and f (0) = 0 then the value of f (1) is:$ (1) $- \frac{1}{2}$ (2) $- \frac{1}{4}$ (3) $\frac{1}{2}$ (4) $\frac{1}{4}$ 274. $If \frac{x}{( x - 1 ) ( x ^{2} + 1 ) ^{2}} = \frac{1}{4} (\frac{1}{x - 1} - \frac{x + 1}{x ^{2} + 1}) + y, then y =$ (1) $\frac{1}{2} (\frac{1 - x}{( x ^{2} + 1 ) ^{2}})$ (2) $\frac{1 + x}{3 ( x ^{2} + 1 ) ^{2}}$ (3) $\frac{1 - x}{( x ^{2} - 1 ) ^{2}}$ (4) $\frac{1 + x}{( x ^{2} + 1 ) ^{2}}$ 275. $\int \frac{dx}{x x ^{6} - 16} =$ (1) $\frac{1}{3} sec^{- 1} (\frac{x ^{3}}{4}) + c$ (2) $cosh^{- 1} (\frac{x ^{3}}{4}) + c$ (3) $\frac{1}{12} sec^{- 1} (\frac{x ^{3}}{4}) + c$ (4) $sec^{- 1} (\frac{x ^{3}}{4}) + c$ 276. $If f (x) = \int_{- 1}^{x} \| t \| dt, then for any x \geq 0, f (x) =$ (1) $1 + x^{2}$ (2) $\frac{1}{2} (1 - x^{2})$ (3) $1 - x^{2}$ (4) $\frac{1}{2} (1 + x^{2})$ 277. $\sum (2 x - 3) 3 x + 2 dx =$ (1) $\frac{2}{135} (54 x^{2} - 123 x + 106) 3 x + 2 + c$ (2) $\frac{2}{135} (54 x^{2} + 123 x - 106) 3 x + 2 + c$ (3) $\frac{2}{135} (54 x^{2} - 123 x - 106) 3 x + 2 + c$ (4) $\frac{2}{135} (54 x^{2} - 195 x - 106) 3 x + 2 + c$ 278. $\int e^{x} \cdot x^{5} dx is$ (1) $e^{x} (x^{5} - 5 x^{4} + 20 x^{3} - 60 x^{2} + 120 x - 120) + C$ (2) $e^{x} (x^{5} + 5 x^{4} + 20 x^{3} - 60 x^{2} - 120 x + 120) + C$ (3) $e^{x} (x^{5} + 5 x^{4} + 20 x^{3} + 60 x^{2} + 120 x + 120) + C$ (4) $e^{x} (x^{5} - 5 x^{4} - 20 x^{3} - 60 x^{2} - 120 x - 120) + C$ 279. $\int_{0}^{π /2} \frac{sin ^{1000} xdx}{sin ^{1000} x + cos ^{1000} x} is equal to$ (1) $1000$ (2) $1$ (3) $π /2$ (4) $π /4$ 280. $\int \frac{sin 2 x ( 1 - \frac{3}{2} cos x )}{e ^{sin^{2} x + cos^{3} x}} dx =$ (1) $e^{sin^{2} x + cos^{3} x} + c, where c is a constant of integration.$ (2) $- e^{- (sin^{2} x + cos^{3} x)} + c, where c is a constant of integration.$ (3) $e^{- (sin^{2} x + cos^{3} x)^{2}} + c, where c is a constant of integration.$ (4) $e^{sin^{2} x + cos x} + c, where c is a constant of integration.$ 281. $If \int \frac{( 2 x + 3 )}{x ( x + 1 ) ( x + 2 ) ( x + 3 ) + 1} dx = - \frac{1}{p x ^{2} + qx + r} + c, then \frac{3 p - q}{r} =$ (1) $0$ (2) $1$ (3) $2$ (4) $-1$ 282. $If tan α = \frac{4}{3}, then \int \frac{1}{3 cos x - 4 sin x} dx =$ (1) $\frac{1}{5} log (tan (\frac{x}{2} + \frac{α}{2})) + c$ (2) $\frac{1}{5} log (tan (\frac{π}{4} + \frac{x}{2} + \frac{α}{2})) + c$ (3) $\frac{1}{5} log (tan (\frac{π}{4} - \frac{x}{2} - \frac{α}{2})) + c$ (4) $\frac{1}{5} log \| tan (sec x + tan x) \| + c$ 283. $If f (x) = sin^{6} x + cos^{6} x + 2 sin^{3} x cos^{3} x, then \int_{0}^{π /4} \frac{sin ^{2} 2 x}{f ( x )} dx =$ (1) $2$ (2) $\frac{2}{3}$ (3) $\frac{- 2}{3}$ (4) $\frac{1}{6}$ 284. $If f (x) = \int \frac{x ^{2} + sin ^{2} x}{1 + x ^{2}} \cdot sec^{2} xdx and f (0) = 0, then f (1) =$ (1) $\frac{π}{4} - 1$ (2) $1 - \frac{π}{4}$ (3) $tan 1 + \frac{π}{4}$ (4) $tan 1 - \frac{π}{4}$ 285. $\int \frac{sin α}{1 + cos α} dα is equal to$ (1) $- 22 cos (\frac{α}{2}) + C$ (2) $22 cos (\frac{α}{2}) + C$ (3) $2 cos (\frac{α}{2}) + C$ (4) $- 2 cos (\frac{α}{2}) + C$ 286. $\sum \frac{3 ^{x}}{9 ^{x} - 1} dx =$ (1) $\frac{1}{l o g 3} lo g (3^{x} + 9^{x} - 1) + c$ (2) $\frac{1}{l o g 3} lo g (3^{x} - 9^{x} - 1) + c$ (3) $\frac{1}{l o g 9} lo g (3^{x} - 9^{x} - 1) + c$ (4) $\frac{1}{l o g 9} lo g (9^{x} - 9^{x} - 1) + c$ 287. $Let f (x) = \int \frac{x}{( 1 + x ) ^{2}} dx (x \geq 0) Then f (3) - f (1) is equal to :$ (1) $\frac{π}{12} + \frac{1}{2} - \frac{3}{4}$ (2) $- \frac{π}{6} + \frac{1}{2} + \frac{3}{4}$ (3) $- \frac{π}{12} + \frac{1}{2} + \frac{3}{4}$ (4) $\frac{π}{6} + \frac{1}{2} - \frac{3}{4}$ 288. $Let x \neq = \frac{- 3}{5}, \frac{2}{5}, if f (\frac{2 x + 1}{5 x + 3}) = x + 2, then \int f (x) dx =$ (1) $\frac{7}{5} x - \frac{1}{5} lo g ∣5 x + 3∣ + c$ (2) $\frac{7}{5} x - \frac{1}{25} lo g ∣5 x + 3∣ + c$ (3) $\frac{7}{5} x - \frac{1}{25} lo g ∣5 x - 2∣ + c$ (4) $\frac{7}{5} x - \frac{1}{5} lo g ∣5 x - 2∣ + c$ 289. $[.] represents a greatest integer function. If \int_{3}^{18} [x] dx = a + b 2 + c 3, then a + b + c =$ (1) $0$ (2) $1$ (3) $-1$ (4) $2$ 290. $If x \neq = (2 n + 1) \frac{π}{2}, then \int \frac{cos ^{3} x}{( 1 + sin x ) ^{4}} dx =$ (1) $- \frac{cos ^{4} x}{( 1 + sin x ) ^{3}} + c$ (2) $- \frac{cos ^{3} x}{( 1 + sin x ) ^{3}} + c$ (3) $- \frac{cos ^{4} x}{( 1 + sin x ) ^{4}} + c$ (4) $- \frac{cos ^{4} x}{4 ( 1 + sin x ) ^{4}} + c$ 291. $Let g (x) be the anti-derivative of f (x) . Then the function for which log_{e} (1 + (g (x))^{2}) + c is an anti-derivative is$ (1) $(1 + (g (x))^{2}) g^{’} (x) f (x)$ (2) $\frac{- 2 f ( x ) g ( x )}{1 + g ( x )}$ (3) $\frac{2 f ( x ) g ( x )}{1 + ( g ( x ) ) ^{2}}$ (4) $\frac{2 g ( x )}{1 + ( g ( x ) ) ^{2}}$ 292. $If \int \frac{dx}{16 - 9 x ^{2}} = A sin - 1 (B x) + C, then A + B =$ (1) $\frac{9}{4}$ (2) $\frac{19}{4}$ (3) $\frac{3}{4}$ (4) $\frac{13}{12}$ 293. $If \int e^{x} (\frac{x + 2}{x + 4})^{2} dx = f (x) + c arbitrary constant, then f (x) =$ (1) $\frac{x e ^{x}}{x + 4}$ (2) $\frac{e ^{x}}{x + 4}$ (3) $\frac{x e ^{x}}{( x + 4 ) ^{2}}$ (4) $\frac{e ^{x}}{( x + 4 ) ^{2}}$ 294. $lim n \to \infty (\frac{n}{n ^{2} + 1} + \frac{n}{2 ^{2} + n ^{2}} + \dots + \frac{1}{2 n}) =$ (1) $\frac{π}{4}$ (2) $log 2$ (3) $0$ (4) $1$ 295. $If \frac{1}{x ^{4} + x ^{2} + 1} = \frac{Ax + B}{x ^{2} + x + 1} + \frac{Cx + D}{x ^{2} - x + 1}, then cos^{- 1} (A + B + C + D) =$ (1) $\frac{π}{2}$ (2) $0$ (3) $\frac{π}{6}$ (4) $\frac{π}{3}$ 296. $The value of \int \frac{e ^{x} ( 1 + x ) dx}{cos ^{2} ( e ^{x} \cdot x )} is equal to$ (1) $- cot (e^{x} \cdot x) + c$ (2) $tan (e^{x} \cdot x) + c$ (3) $tan (e^{x}) + c$ (4) $cot (e^{x}) + c$ 297. $0 < x < 1, \sum \frac{dx}{x ^{2} - x ^{5}} = \frac{1}{3} log \| f (x) \| + C, then f (\frac{1}{2}) =$ (1) $\frac{( 8 - 7 )}{( 8 + 7 )}$ (2) $\frac{( 8 + 7 )}{( 8 - 7 )}$ (3) $2 (8 - 7)$ (4) $2 (8 - 7)^{2}$ 298. $If \int \frac{x + 3}{( x - 1 ) ^{2} ( 2 x - 1 )} dx = \frac{A}{x - 1} + Blog (2 x - 1) + Clog (x - 1) + K then A + B + C =$ (1) $3$ (2) $11$ (3) $-4$ (4) $-11$ 299. $If I_{n} = \int \frac{sin nx}{sin x} dx for n = 1, 2, 3, \dots, then I_{6} =$ (1) $\frac{3}{5} sin 3 x + \frac{8}{5} sin 5 x - sin x + c$ (2) $\frac{2}{5} sin 5 x - \frac{5}{3} sin 3 x - 2 sin x + c$ (3) $\frac{2}{3} sin 5 x - \frac{8}{3} sin 5 x + 4 sin x + c$ (4) $\frac{2}{5} sin 5 x - \frac{8}{3} sin^{3} x + 4 sin x + c$ 300. $\int \frac{cosec x}{cos ^{2} ( 1 + log tan \frac{x}{2} )} dx =$ (1) $sin^{2} (1 + log tan \frac{x}{2}) + c$ (2) $tan (1 + log tan \frac{x}{2}) + c$ (3) $sec^{2} (1 + log tan \frac{x}{2}) + c$ (4) $- tan (1 + log tan \frac{x}{2}) + c$