Indefinite Integration PYQs Set 14 (PDF)

Set 14 of 16 · Questions 391-420 of 459

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Indefinite Integration · Set 14 (Q391-420)
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391. $\int_{0}^{π /4} \frac{d x}{c o s ^{3} x \cdot 2 s i n 2 x} =$ (1) $\frac{6}{5}$ (2) $\frac{3}{5}$ (3) $\frac{4}{5}$ (4) $\frac{8}{5}$ 392. $\int_{- π}^{π} cos m x cos n x d x + \int_{- π}^{π} sin m x cos n x d x =$ (1) $0 , if m = n and m, n \in Z$ (2) $π if m = n, m, n \in Z$ (3) $π if m \neq = n, m, n \in Z$ (4) $2 π \forall m, n \in R$ 393. $\int_{0}^{π} \frac{x c o s ^{2} x}{1 + s i n x} d x =$ (1) $\frac{π ( π - 2 )}{2}$ (2) $1$ (3) $\frac{π ( x + 2 )}{2}$ (4) $\frac{π}{4}$ 394. $\int tan^{- 1} (1 - x + x^{2}) d x + \int tan^{- 1} (x) d x + \int tan^{- 1} (1 - x) d x =$ (1) $\frac{π}{2} x + C$ (2) $\frac{π}{4} x + C$ (3) $x + C$ (4) $πx + C$ 395. $[\cdot] is the greatest integer function then \int_{0}^{2 π} [∣ sin x ∣ + ∣ cos x ∣] d x =$ (1) $\frac{π}{2}$ (2) $π$ (3) $\frac{3 π}{2}$ (4) $2 π$ 396. $Let I = \int_{- π /4}^{π /4} \frac{1}{2 - c o s 2 x} (\frac{3}{π} + lo g (\frac{4 + s i n x}{4 - s i n x})) d x . Given that \int \frac{d x}{1 + k x ^{2}} = \frac{1}{k} Tan^{- 1} (k x) + c, Tan^{- 1} (0) = 0, Tan^{- 1} (3) = \frac{π}{3} . Then 3 I^{2} =$ (1) $4$ (2) $9$ (3) $16$ (4) $1$ 397. $\int_{π /4}^{π /2} \frac{3 d x}{1 + e ^{8 s i n (x - \frac{3 π}{8})}} =$ (1) $\frac{3 2}{4} π$ (2) $\frac{3}{4} π$ (3) $\frac{π}{8}$ (4) $\frac{3}{8} π$ 398. $\int_{0}^{π /2} \frac{s i n ^{3} x c o s x}{s i n ^{4} x + c o s ^{4} x} d x =$ (1) $π$ (2) $\frac{π}{2}$ (3) $\frac{π}{4}$ (4) $\frac{π}{8}$ 399. $lim_{n \to \infty} \sum_{k = 1}^{n} \frac{k}{n ^{2} + k ^{2}} =$ (1) $\frac{1}{2} lo g 2$ (2) $2 log 2$ (3) $\frac{1}{3} lo g 2$ (4) $3 log 2$ 400. $Let n \geq 2 be a natural number and 0 < θ < \frac{π}{2} . Then \int \frac{( s i n ^{n} θ - s i n θ ) ^{\frac{1}{n}} c o s θ}{s i n ^{n + 1} θ} d θ is equal to:$ (1) $\frac{n}{n ^{2} - 1} (1 - \frac{1}{sin ^{n - 1} θ})^{\frac{n + 1}{n}} + C$ (2) $\frac{n}{n ^{2} + 1} (1 - \frac{1}{sin ^{n - 1} θ})^{\frac{n + 1}{n}} + C$ (3) $\frac{n}{n ^{2} - 1} (1 + \frac{1}{sin ^{n - 1} θ})^{\frac{n + 1}{n}} + C$ (4) $\frac{n}{n ^{2} - 1} (1 - \frac{1}{sin ^{n + 1} θ})^{\frac{n + 1}{n}} + C (where c is a constant of integration)$ 401. $\int_{0}^{3} (sin (\frac{π}{3} x) - cos (\frac{π}{3} x)) d x =$ (1) $\frac{- 6}{π}$ (2) $0$ (3) $\frac{- 3}{π}$ (4) $\frac{6}{π}$ 402. $\int_{0}^{π /2} sin^{4} θ cos^{3} θ d θ =$ (1) $\frac{1}{35}$ (2) $\frac{2}{35}$ (3) $\frac{4}{35}$ (4) $\frac{8}{35}$ 403. $\int_{0}^{10} (5 - 10 x - x^{2}) d x =$ (1) $50 - 25 π$ (2) $(100 - 25 π)$ (3) $\frac{1}{2} (100 - 25 π)$ (4) $\frac{1}{4} (100 - 25 π)$ 404. $Match the following List I I. \int_{- 1}^{1} x ∣ x ∣ d x II. \int_{0}^{π /2} (1 + lo g \frac{4 + 3 s i n x}{4 + 3 c o s x}) d x III. \int_{0}^{a} f (x) d x IV. \int_{- a}^{a} f (x) d x List II (A) \frac{π}{2} (B) \int_{0}^{a /2} f (x) d x (C) \int_{0}^{a} [f (x) + f (- x)] d x (D) 0 (E) \int_{0}^{a} f (a - x) d x$ (1) $I-(D), II-(A), III-(E), IV-(C)$ (2) $I-(D), II-(A), III-(C), IV-(B)$ (3) $I-(D), II-(C), III-(A), IV-(E)$ (4) $I-(A), II-(D), III-(B), IV-(C)$ 405. $I_{m, n} = \int x^{m} (lo g x)^{n} d x =$ (1) $\frac{x ^{m + 1}}{m + 1} (lo g x)^{n} - \frac{n}{m + 1} I_{m, n - 1}$ (2) $\frac{x ^{m}}{m} (lo g x)^{n} - \frac{n - 1}{m + 1} I_{m + 1, n - 1}$ (3) $\frac{x ^{m + 1}}{m} \cdot \frac{( l o g x ) ^{n + 1}}{n + 1} - \frac{n}{m + 1} I_{m, n - 1}$ (4) $\frac{x ^{m} ( l o g x ) ^{n + 1}}{n + 1} - \frac{n}{m + 1} I_{m, n - 1}$ 406. $\int \frac{d x}{( x - 1 ) x + 2} =$ (1) $\frac{2}{3} log (\frac{( x + 2 ) + 3}{( x + 2 ) - 3}) + c$ (2) $\frac{- 1}{3} log (\frac{( x + 2 ) - 3}{( x + 2 ) + 3}) + c$ (3) $\frac{1}{3} log (\frac{( x + 2 ) + 3}{( x + 2 ) - 3}) + c$ (4) $\frac{1}{3} log (\frac{( x + 2 ) - 3}{( x + 2 ) + 3}) + c$ 407. $\int x^{4} e^{2 x} d x =$ (1) $\frac{e ^{2 x}}{4} (2 x^{4} - 4 x^{3} + 6 x^{2} - 6 x + 3) + C$ (2) $\frac{e ^{2 x}}{2} (2 x^{4} - 4 x^{3} + 6 x^{2} - 6 x + 3) + C$ (3) $\frac{e ^{2 x}}{8} (2 x^{4} + 4 x^{3} + 6 x^{2} + 6 x + 3) + C$ (4) $- \frac{e ^{2 x}}{4} (2 x^{4} + 4 x^{3} + 6 x^{2} + 6 x + 3) + C$ 408. $\int x + x^{2} + 2 d x =$ (1) $\frac{3}{2} (x + x + 2)^{3/2} - 2 (x + x^{2} + 2)^{1/4} + C$ (2) $\frac{1}{3} (x + x^{2} + 2)^{3/2} - 2 (x + x^{2} + 2)^{1/4} + C$ (3) $(x + x^{2} + 2)^{- 3/2} - 2 (x + x^{2} + 2)^{- 1/2} + C$ (4) $\frac{( x + x ^{2} + 2 ) ^{2} - 6}{3 x + x ^{2} + 2} + C$ 409. $\int \frac{x + s i n x}{1 + c o s x} d x =$ (1) $xtan \frac{x}{2} + C$ (2) $xsin \frac{x}{2} + cos \frac{x}{2} + C$ (3) $xtan \frac{x}{2} + sec \frac{x}{2} + C$ (4) $xsec \frac{x}{2} + tan \frac{x}{2} + C$ 410. $\int \frac{d x}{( x ^{2} + 1 ) ( x ^{2} + 4 )} =$ (1) $\frac{1}{3} tan^{- 1} x + \frac{1}{6} tan^{- 1} \frac{x}{2} + c$ (2) $\frac{1}{3} tan^{- 1} x - \frac{1}{3} tan^{- 1} \frac{x}{2} + c$ (3) $\frac{1}{3} tan^{- 1} x + \frac{1}{3} tan^{- 1} \frac{x}{2} + c$ (4) $\frac{1}{3} tan^{- 1} x - \frac{1}{6} tan^{- 1} \frac{x}{2} + c$ 411. $\int \frac{d x}{s i n x + s i n 2 x} =$ (1) $\frac{1}{2} log_{e} \| 1 + cos x \| + \frac{1}{6} log_{e} \| 1 - cos x \| - \frac{2}{3} log_{e} \| 1 + 2 cos x \| + c$ (2) $\frac{1}{2} log_{e} \| 1 + cos x \| - \frac{2}{3} log_{e} \| 1 - cos x \| + \frac{1}{2} log_{e} \| 1 + 2 cos x \| + c$ (3) $\frac{1}{2} log_{e} \| 1 + sin x \| - \frac{1}{3} log_{e} \| 1 - sin x \| - \frac{1}{3} log_{e} \| 1 + cos x \| + c$ (4) $\frac{1}{3} log_{e} \| 1 - sin x \| + \frac{1}{2} log_{e} \| 1 + cos x \| - \frac{2}{3} log_{e} \| 1 - 2 cos x \| + c$ 412. $\int \frac{t a n ^{- 1} x}{x ^{3}} d x =$ (1) $- (\frac{x}{2} + \frac{1}{2}) tan^{- 1} x - \frac{1}{2 x} + C$ (2) $- \frac{x}{2} + \frac{1}{2} (\frac{1}{x ^{2}} + 1) tan^{- 1} x - \frac{1}{2 x ^{2}} + C$ (3) $- \frac{1}{2 x} - (\frac{1}{2} + \frac{1}{2 x ^{2}}) tan^{- 1} x + C$ (4) $(\frac{1}{2 x} + \frac{1}{2 x ^{2}}) tan^{- 1} x + C$ 413. $\int 5 - 2 x + x^{2} dx =$ (1) $\frac{x - 1}{2} 5 + 2 x + x^{2} + 2 lo g [(x - 1) + 5 + 2 x + x^{2}] + C$ (2) $\frac{x - 1}{2} 5 - 2 x + x^{2} + 2 lo g [(x + 1) + x^{2} + 2 x + 5] + C$ (3) $\frac{x - 1}{2} 5 - 2 x + x^{2} + 2 lo g [(x - 1) + 5 - 2 x + x^{2}] + C$ (4) $\frac{x}{2} 5 - 2 x + x^{2} + 4 lo g [(x + 1) + x^{2} - 2 x + 5] + C$ 414. $Let \frac{1}{( x ^{2} - 3 ) ^{2}} = \frac{A _{1}}{x - 3} + \frac{A _{2}}{( x - 3 ) ^{2}} + \frac{A _{3}}{x + 3} + \frac{A _{4}}{( x + 3 ) ^{2}} Then, consider the following statements. (i) All the A_{i}'s are not distinct. (ii) There exists a pair, A_{p} and A_{q} such that A_{p}^{2} = A_{q}^{2} (p \neq = q) (iii) \sum_{i = 1}^{4} A_{i} = \frac{1}{6} (iv) \sum_{i = 1}^{4} A_{i} = 1 Which one of the following is true?$ (1) $Only statement (iii) is false$ (2) $Both the statements (ii) and (iv) are false$ (3) $Only statement (iv) is false$ (4) $Both the statements (i) and (iii) are false$ 415. $\int (lo g x)^{2} d x =$ (1) $xlog x - 2 xlog x + c$ (2) $xlog x + 2 xlog x + c$ (3) $x (log x)^{2} - 2 x (log x - 1) + c$ (4) $x (log x)^{2} + 2 x (log x - 1) + c$ 416. $\int \frac{d x}{s i n ( x - a ) c o s ( x - b )} =$ (1) $\frac{1}{sin ( a - b )} log (\frac{sin ( x - a )}{cos ( x - b )}) + C$ (2) $\frac{1}{cos ( b - a )} log (\frac{sin ( x - a )}{cos ( x - b )}) + C$ (3) $\frac{1}{cos ( b - a )} [log \| sin (x - a) cos (x - b) \|] + C$ (4) $\frac{1}{sin ( a - b )} [log \| sin (x - a) cos (x - b) \|] + C$ 417. $\int \frac{1}{16 - 7 s i n ^{2} x} d x =$ (1) $\frac{1}{12} Tan^{- 1} (\frac{3 tan x}{4}) + c$ (2) $\frac{1}{3} Sin^{- 1} (\frac{3 sin x}{4}) + c$ (3) $\frac{1}{12} log (\frac{4 - 7 sin x}{4 + 7 sin x}) + c$ (4) $\frac{1}{12} log (\frac{4 + 7 sin x}{4 - 7 sin x}) + c$ 418. $\int \frac{dx}{x ( x ^{4} + 1 )} =$ (1) $\frac{1}{4} log (\frac{x ^{4} + 1}{x ^{4}}) + C$ (2) $\frac{1}{4} log (\frac{x ^{4}}{x ^{4} + 1}) + C$ (3) $\frac{1}{4} log (x^{4} + 1) + C$ (4) $\frac{1}{4} log (\frac{x ^{4}}{x ^{4} + 2}) + C$ 419. $\int e^{- 3 x} (x^{2} + sin 4 x) d x =$ (1) $- e^{- 3 x} (\frac{x ^{2}}{3} + \frac{2 x}{9} + \frac{2}{27} + \frac{3}{25} sin 4 x + \frac{4}{25} cos 4 x) + C$ (2) $- e^{- 3 x} (\frac{x ^{2}}{3} - \frac{2 x}{9} + \frac{2}{27} + \frac{3}{25} sin 4 x + \frac{4}{25} cos 4 x) + C$ (3) $- e^{- 3 x} (\frac{x ^{2}}{3} + \frac{2 x}{9} + \frac{2}{27} + \frac{3}{25} sin 4 x - \frac{4}{25} cos 4 x) + C$ (4) $- e^{- 3 x} (\frac{x ^{2}}{3} - \frac{2 x}{9} + \frac{2}{27} + \frac{3}{25} sin 4 x - \frac{4}{25} cos 4 x) + C$ 420. $For real numbers α, β, γ and δ, if \int \frac{( x ^{2} - 1 ) + tan ^{- 1} ( \frac{x ^{2} + 1}{x} )}{( x ^{4} + 3 x ^{2} + 1 ) tan ^{- 1} ( \frac{x ^{2} + 1}{x} )} dx = α log_{e} (tan^{- 1} (\frac{x ^{2} + 1}{x})) + β tan^{- 1} (\frac{γ ( x ^{2} - 1 )}{x}) + δ tan^{- 1} (\frac{x ^{2} + 1}{x}) + C where C is an arbitrary constant, then the value is 10 (α + βγ + δ) equal to_____$