Indefinite Integration PYQs Set 16 (PDF)

Set 16 of 16 · Questions 451-459 of 459

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Indefinite Integration · Set 16 (Q451-459)
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451. $If f (x) = sin (tan^{- 1} x), then \int_{0}^{1} x f^{’ ’} (x) dx =$ (1) $1 - \frac{3}{2 2}$ (2) $- \frac{1}{2 2}$ (3) $\frac{1}{2}$ (4) $- 2$ 452. $The value of the integral \int_{- π /4}^{π /4} log (sec θ - tan θ) dθ is$ (1) $0$ (2) $\frac{π}{4}$ (3) $π$ (4) $\frac{π}{2}$ 453. $\int \frac{1}{3 - 6 x - 9 x ^{2}} dx is equal to$ (1) $sin^{- 1} (\frac{3 x + 1}{2}) + c$ (2) $sin^{- 1} (\frac{3 x + 1}{6}) + c$ (3) $\frac{1}{3} sin^{- 1} (\frac{3 x + 1}{2}) + c$ (4) $sin^{- 1} (\frac{2 x + 1}{3}) + c$ 454. $\int_{2}^{4} \| x-2 \| + \| x-3 \| dx is equal to$ (1) $1$ (2) $2$ (3) $3$ (4) $4$ 455. $Let I (x) = \int \frac{x + 1}{x ( 1 + x e ^{x} ) ^{2}} d x, x > 0 . If lim_{x \to \infty} I_{(} x_{)} = 0 then I(a) is equal to$ (1) $\frac{e + 2}{e + 1} - lo g_{e} (e + 1)$ (2) $\frac{e + 1}{e + 2} + lo g_{e} (e + 1)$ (3) $\frac{e + 1}{e + 2} - lo g_{e} (e + 1)$ (4) $\frac{e + 2}{e + 1} + lo g_{e} (e + 1)$ 456. $The integral \int \frac{1}{4 ( x - 1 ) ^{3} ( x + 2 ) ^{5}} d x is equal to : (where is C a constant of integration)$ (1) $\frac{3}{4} (\frac{x + 2}{x - 1})^{\frac{1}{4}} + C$ (2) $\frac{3}{4} (\frac{x + 2}{x - 1})^{\frac{5}{4}} + C$ (3) $\frac{4}{3} (\frac{x - 1}{x - 2})^{\frac{1}{4}} + C$ (4) $\frac{4}{3} (\frac{x - 1}{x + 2})^{\frac{5}{4}} + C$ 457. $\int_{l o g_{e} 2}^{x} \frac{d t}{e ^{t} - 1} = \frac{π}{6} \Rightarrow x =$ (1) $2 \cdot loge 2$ (2) $3 \cdot loge 2$ (3) $4 \cdot loge 2$ (4) $8 \cdot log e 2$ 458. $\int_{0}^{π /4} log (\frac{sin x + cos x}{cos x}) dx is equal to$ (1) $\frac{π}{2} lo g 2$ (2) $log 2$ (3) $\frac{π}{4} lo g 2$ (4) $\frac{π}{8} lo g 2$ 459. $The integral \int \frac{( 1 - \frac{1}{3} ) ( c o s x - s i n x )}{( 1 + \frac{2}{3} s i n 2 x )} d x is equal to$ (1) $\frac{1}{2} lo g_{e} \frac{t a n ( \frac{x}{2} + \frac{π}{12} )}{( \frac{x}{2} + \frac{π}{6} )} + C$ (2) $\frac{1}{2} lo g_{e} \frac{t a n ( \frac{x}{2} + \frac{π}{6} )}{( \frac{x}{2} + \frac{π}{3} )} + C$ (3) $lo g_{e} \frac{t a n ( \frac{x}{2} + \frac{π}{6} )}{t a n ( \frac{x}{2} + \frac{π}{12} )} + C$ (4) $\frac{1}{2} lo g_{e} \frac{t a n ( \frac{x}{2} - \frac{π}{12} )}{t a n ( \frac{x}{2} - \frac{π}{6} )} + C$