Indefinite Integration PYQs Set 15 (PDF)

Set 15 of 16 · Questions 421-450 of 459

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Indefinite Integration · Set 15 (Q421-450)
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421. $\int \frac{d x}{( 2 a x + x ^{2} ) ^{3/2}} =$ (1) $\frac{- 1}{a ^{2}} \frac{( x + a )}{2 ax + x ^{2}} + c$ (2) $\frac{- ( x + a )}{2 ax + x ^{2}} + c$ (3) $\frac{1}{2 a ^{2}} \frac{( x + a )}{2 ax + x ^{2}} + c$ (4) $\frac{- 1}{a} \frac{( x + a )}{2 ax + x ^{2}} + c$ 422. $\frac{x ^{4}}{x ^{3} - 3 x + 2} is a$ (1) $proper fraction$ (2) $improper fraction$ (3) $mixed fraction$ (4) $not a fraction$ 423. $Let F(x) be an indefinite integral of sin^{2} x . STATEMENT-1 : The function F(x) satisfies F (x + π) = F (x) for all real x. because STATEMENT-2 : sin^{2} (x + π) = sin^{2} x for all real x.$ (1) $Statement-1 is True, Statement-2 is True ; Statement-2 is a correct explanation for Statement-1$ (2) $Statement-1 is True, Statement-2 is True ; Statement-2 is NOT a correct explanation for Statement-1$ (3) $Statement-1 is True, Statement-2 is False$ (4) $Statement-1 is False, Statement-2 is True$ 424. $The integral \int \frac{d x}{x ^{2} ( x ^{4} + 1 ) ^{3/4}} equals$ (1) $+ c$ (2) $(x4 + 1)1/4 + c$ (3) $-(x4 + 1)1/4 + c$ (4) $-+ c$ 425. $\int_{0}^{2 a} f (x) d x =$ (1) $2 \int_{0}^{a} f (x) d x$ (2) $\int_{0}^{a} (f (x) + f (x + a)) d x$ (3) $0$ (4) $\int_{0}^{2 a} f (2 a + x) d x$ 426. $is equal to$ (1) $- \frac{x - 1}{x + 1} + C$ (2) $\frac{x - 1}{x ^{2} + 1} + C$ (3) $- \frac{x + 1}{x - 1} + C$ (4) $\frac{x ^{2} + 1}{x - 1} + C$ 427. $\int \frac{s i n ^{8} x - c o s ^{8} x}{1 - 2 s i n ^{2} x c o s ^{2} x} d x =$ (1) $\frac{1}{2} cos 2 x + c$ (2) $\frac{- 1}{2} cos 2 x + c$ (3) $\frac{- 1}{( 1 + t a n x ) ^{2}} + c$ (4) $\frac{- 1}{2} sin 2 x + c$ 428. $If \int_{0}^{1} \frac{1}{( 5 + 2 x - 2 x ^{2} ) ( 1 + e ^{(2 - 4 x)} )} d x = \frac{1}{α} lo g_{e} (\frac{α + 1}{β}), α, β > 0 then α^{4} - β^{4} equal to$ (1) $19$ (2) $-21$ (3) $0$ (4) $21$ 429. $If for x, y \in R, x > 0 y = l o g_{10} x + l o g_{10} x^{1/3} + l o g_{10} x^{1/9} + ... upto \infty terms and \frac{2 + 4 + 6 + ... + 2 y}{3 + 6 + 9 + .... + 3 y} = \frac{4}{l o g _{10} x}, then the ordered pair (x,y) is equal to :$ (1) $(1 0^{6},6)$ (2) $(1 0^{4},6)$ (3) $(1 0^{2},3)$ (4) $(1 0^{6},9$ 430. $\int \frac{( x ^{2} + 1 ) e ^{x}}{( x + 1 ) ^{2}} d x = f (x) e^{x} + C, Where C constant, then \frac{d ^{3} f}{d x ^{2}} at x = 1 is equal to :$ (1) $- \frac{3}{4}$ (2) $\frac{3}{4}$ (3) $- \frac{3}{2}$ (4) $\frac{3}{2}$ 431. $Let n \geq 2 be a natural number and 0 < θ < π / 2. Then \int \frac{( s i n ^{n} θ - s i n θ ) ^{\frac{1}{n}} c o s θ}{s i n ^{n + 1} θ} dθ is equal to : (where C is a constant of integration)$ (1) $+ C$ (2) $+ C$ (3) $+ C$ (4) $+ C$ 432. $\int_{0}^{π /2} sin^{6} x cos^{4} x d x =$ (1) $\frac{π}{256}$ (2) $\frac{π}{512}$ (3) $\frac{3 π}{512}$ (4) $\frac{5 π}{512}$ 433. $\int_{0}^{π /2} x^{3} sin x d x =$ (1) $\frac{3 π ^{2}}{4} - 3 π + 6$ (2) $\frac{3 π ^{2}}{4} + 3 π - 6$ (3) $\frac{3 π ^{2}}{4} + 6$ (4) $\frac{3 π ^{2}}{4} - 6$ 434. $\int_{0}^{1} x (1 - x)^{3/2} d x =$ (1) $\frac{4}{35}$ (2) $\frac{- 2}{35}$ (3) $\frac{- 8}{35}$ (4) $\frac{24}{35}$ 435. $\int_{0}^{2} x^{5/2} 2 - x d x =$ (1) $\frac{5 π}{16}$ (2) $\frac{5}{4}$ (3) $\frac{5 π}{8}$ (4) $\frac{5}{8}$ 436. $\int_{0}^{1} a^{k} x^{k} d x =$ (1) $lim_{n \to \infty} \frac{a ^{k} ( 1 + 2 ^{k} + 3 ^{k} \dots + n ^{k} )}{n ^{k + 1}}$ (2) $lim_{n \to \infty} \frac{a ^{k} + a ^{k} + \dots + a ^{k}}{n ^{k + 1}}$ (3) $lim_{n \to \infty} \frac{1}{n} \sum (\frac{r}{n})^{k}$ (4) $lim_{n \to \infty} \frac{1}{n} \sum (\frac{2 r}{n})^{k}$ 437. $\int_{0}^{2} [x^{2}] d x =$ (1) $5 - 2 - 3$ (2) $5 - 2 + 3$ (3) $5 + 2 - 3$ (4) $- 5 - 2 - 3$ 438. $\int_{- π}^{π} x^{2} sin x d x =$ (1) $π^{2}$ (2) $\frac{π ^{2}}{2}$ (3) $0$ (4) $2 π^{2}$ 439. $\int_{1}^{e^{2}} \frac{d x}{x ( 1 + l o g x ) ^{2}} =$ (1) $\frac{2}{3}$ (2) $\frac{1}{3}$ (3) $\frac{3}{2}$ (4) $log 2$ 440. $\int_{1}^{2} x 4 - x^{2} d x =$ (1) $3$ (2) $2$ (3) $\frac{1}{3}$ (4) $\frac{1}{2}$ 441. $\int_{- 1}^{2} ∣ x ∣ d x =$ (1) $1$ (2) $2$ (3) $\frac{5}{2}$ (4) $\frac{3}{2}$ 442. $\int_{3}^{5} (x - 3)^{3} (5 - x)^{5} d x =$ (1) $\frac{64}{63}$ (2) $\frac{25}{7}$ (3) $\frac{3}{32}$ (4) $\frac{16}{25}$ 443. $Let f (x) = \int ex (x - 1) (x - 2) d x then f decreases in the interval:$ (1) $(- \infty, 2)$ (2) $(- 2, - 1)$ (3) $(1, 2)$ (4) $(2, + \infty)$ 444. $If the integral \int \frac{5 t a n x}{t a n x - 2} dx = x + a λn \|sin x - 2 cos x\| + k, then a is equal to :$ (1) $- 1$ (2) $- 2$ (3) $1$ (4) $2$ 445. $\int_{0}^{π /4} tan^{2} (x) d x =$ (1) $\frac{π}{4}$ (2) $\frac{π}{4} - 1$ (3) $1 - \frac{π}{4}$ (4) $0$ 446. $The value of \frac{e ^{- \frac{π}{4}} + \int _{0}^{\frac{π}{4}} e ^{- x} t a n ^{50} x d x}{\int _{0}^{\frac{π}{4}} e ^{- x} ( t a n ^{49} x + t a n ^{51} x ) d x}$ (1) $51$ (2) $50$ (3) $25$ (4) $49$ 447. $The integral \int ((\frac{x}{2})^{x} + (\frac{2}{x})^{x}) lo g_{2} x d x is equal to$ (1) $(\frac{x}{2})^{x} + (\frac{2}{x})^{x} + C$ (2) $(\frac{x}{2})^{x} - (\frac{2}{x})^{x} + C$ (3) $(\frac{x}{2})^{x} lo g_{2} (\frac{x}{2}) + C$ (4) $(\frac{x}{2})^{x} lo g_{2} (\frac{2}{x}) + C$ 448. $\int_{- 1}^{1} \frac{\| x \|}{x} dx is equal to$ (1) $1$ (2) $-1$ (3) $0$ (4) $\frac{1}{2}$ 449. $If A_{n} = \int_{\frac{π}{2}}^{\infty} e^{- x} \cdot cos^{n} xdx, then \frac{A _{4} - A _{6}}{A _{4}} =$ (1) $\frac{3}{2}$ (2) $\frac{7}{37}$ (3) $\frac{5}{37}$ (4) $\frac{2}{7}$ 450. $The value of \int e^{x} (\frac{x ^{2} + 4 x + 4}{( x + 4 ) ^{2}}) dx is$ (1) $e^{x} (\frac{x}{x + 4}) + c where c is a constant of integration$ (2) $e^{x} (\frac{4}{x + 4}) + c where c is a constant of integration$ (3) $e^{x} (\frac{x}{( x + 4 ) ^{2}}) + c where c is a constant of integration$ (4) $e^{x} (\frac{4}{( x + 4 ) ^{2}}) + c, where c is a constant of integration$