Indefinite Integration PYQs Set 7 (PDF)

Set 7 of 16 · Questions 181-210 of 459

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Indefinite Integration · Set 7 (Q181-210)
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181. $The value of the integral \int_{0}^{π /2} (sin^{100} x - cos^{100} x) dx is$ (1) $\frac{1}{100}$ (2) $\frac{100 !}{( 100 ) ^{100}}$ (3) $\frac{π}{100}$ (4) $0$ 182. $\sum (1 + x) log (1 + x^{2}) dx = (x + \frac{x ^{2}}{2} + \frac{1}{2}) log (1 + x^{2}) + g (x) + C, then g (x) =$ (1) $- 2 x - \frac{x ^{2}}{2} + 2 tan^{- 1} x$ (2) $2 tan^{- 1} x + \frac{x ^{2}}{2} + \frac{x ^{3}}{3}$ (3) $2 tan^{- 1} x - \frac{x ^{2}}{2} + 3 x$ (4) $2 tan^{- 1} x + 3 x + \frac{x ^{3}}{2}$ 183. $\int \frac{1}{sin x \cdot cos ^{2} x} dx =$ (1) $sec x + log \| sec x + tan x \| + c$ (2) $sec x \cdot tan x + c$ (3) $sec x + log \| sec x - tan x \| + c$ (4) $sec x + log \| cosec x - cot x \| + c$ 184. $If I = \int \frac{dx}{sin ( x - a ) sin ( x - b )}, then I is given by$ (1) $\frac{1}{sin ( b - a )} log \| sin (x - a) sin (x - b) \| + c, where c is a constant of integration.$ (2) $log (\frac{sin ( x - a )}{sin ( x - b )}) + c, where c is a constant of integration.$ (3) $\frac{1}{sin ( b - a )} log (\frac{sin ( x - a )}{sin ( x - b )}) + c, where c is a constant of integration.$ (4) $\frac{1}{sin ( b - a )} log (\frac{sin ( x - b )}{sin ( x - a )}) + c, where c is a constant of integration.$ 185. $\int \frac{1}{sin ( x - a ) sin x} dx =$ (1) $sin a (log (sin (x - a) \cdot cosec x)) + c, where c is a constant of integration$ (2) $cosec a (log (sin (x - a) \cdot cosec x)) + c, where c is a constant of integration$ (3) $- sin a (log sin (x - a) \cdot sin x)) + c, where c is a constant of integration.$ (4) $- cosec a (log (sin (x - a) \cdot sin x)) + c, where c is a constant of integration$ 186. $\int \frac{cos 2 x - cos 2 θ}{cos x - cos θ} dx is equal to$ (1) $2 (sin x + x cos θ) + C$ (2) $2 (sin x - x cos θ) + C$ (3) $2 (sin x + 2 x cos θ) + C$ (4) $2 (sin x - 2 x cos θ) + C$ 187. $If f (x) is defined on [- 2, 2] by f (x) = 4 x^{2} - 3 x + 1 and g (x) = \frac{f ( - x ) - f ( x )}{( x ^{2} + 3 )}, then \int_{- 2}^{2} g (x) dx =$ (1) $64$ (2) $-48$ (3) $0$ (4) $24$ 188. $Given that lim_{n \to \infty} \frac{1}{n} \sum_{r = 1}^{np} f (\frac{r}{n}) = \int_{0}^{p} f (x) dx . If f : R \to R is defined by f (x) = x^{2} + 2, then lim n \to \infty \frac{3}{n} (f (\frac{7}{n}) + f (\frac{14}{n}) + f (\frac{21}{n}) + \dots + f (7)) =$ (1) $55$ (2) $57$ (3) $104$ (4) $7$ 189. $If \frac{x ^{2} - x + 1}{( x ^{2} + 1 ) ( x ^{2} + x + 1 )} = \frac{Ax + B}{x ^{2} + 1} + \frac{Cx + D}{x ^{2} + x + 1} then A + 2 B + C + 2 D =$ (1) $0$ (2) $1$ (3) $-1$ (4) $2$ 190. $\int_{0}^{π /4} \frac{sin x + cos x}{7 + 9 sin 2 x} dx is equal to$ (1) $\frac{log 3}{4}$ (2) $\frac{log 3}{36}$ (3) $\frac{log 7}{12}$ (4) $\frac{log 7}{24}$ 191. $If \sum \frac{sin ^{3} x ( tan ^{- 1} ( sec x + cos x ) ) ^{- 1}}{( cos ^{4} x + 3 cos ^{2} x + 1 )} dx = f (x) + C then e^{f (x)} =$ (1) $tan^{- 1} (sec x + cos x)$ (2) $tan (sec x + cos x)$ (3) $\frac{1}{cos ^{4} x + 3 cos ^{2} x + 1}$ (4) $\frac{sin x}{sin ^{3} x + cos ^{4} x + 1}$ 192. $\int \frac{sin 2 x}{sin ^{2} x + 2 cos ^{2} x} dx =$ (1) $- log (1 + sin^{2} x) + C$ (2) $log (1 + cos^{2} x) + C$ (3) $- log (1 + cos^{2} x) + C$ (4) $log (1 + tan^{2} x) + C$ 193. $The value of \int_{- 2}^{2} (a x^{3} + bx + c) dx depends on the$ (1) $value of a$ (2) $value of b$ (3) $values of a and b$ (4) $value of c$ 194. $\sum e^{x} (\frac{2 + sin 2 x}{1 + cos 2 x}) dx =$ (1) $e^{x} sec x + C$ (2) $e^{x} tan x + C$ (3) $e^{x} cot x + C$ (4) $e^{x} cosec x + C$ 195. $let a, b, c be real numbers such that 2 a + 3 b + 6 c = 0 and g (x) \equiv a x^{2} + bx + c = 0 has atleast one root in the interval (1, 2) . If a function f : [1, 2] \to R for which Rolle's mean value theorem holds is such that f (x) is a primitive of g (x), then f (x) =$ (1) $x^{3} - 3 x^{2} + 2 x$ (2) $3 x^{3} - 6 x^{2} + 2 x$ (3) $12 x^{3} - 14 x^{2} + 3 x$ (4) $3 x^{3} - x$ 196. $\sum \frac{x ^{2}}{( x ^{2} - 1 ) ( x ^{2} + 1 )} dx =$ (1) $\frac{1}{4} lo g (\frac{x + 1}{x - 1}) - \frac{1}{2} tan^{- 1} x + c$ (2) $\frac{1}{4} lo g (\frac{x - 1}{x + 1}) + \frac{1}{2} tan^{- 1} x + c$ (3) $\frac{1}{4} lo g (\frac{x - 1}{x + 1}) - \frac{1}{2} tan^{- 1} x + c$ (4) $\frac{1}{4} lo g (\frac{x + 1}{x - 1}) + \frac{1}{2} tan^{- 1} x + c$ 197. $If \frac{2 x ^{2} + 5 x + 6}{( x + 2 ) ^{3}} = \frac{a}{x + 2} + \frac{b}{( x + 2 ) ^{2}} + \frac{c}{( x + 2 ) ^{3}} then a . b + b . c + c . a =$ (1) $28$ (2) $14$ (3) $-10$ (4) $-8$ 198. $If \int \frac{1 - ( cot x ) ^{2021}}{tan x + ( cot x ) ^{2022}} dx = \frac{1}{A} log ((sin x)^{2023} + (cos x)^{2023}) + c, then A is equal to$ (1) $2020$ (2) $2021$ (3) $2022$ (4) $2023$ 199. $\int \frac{x - 1}{( x + 1 ) x ^{3} + x ^{2} + x} dx =$ (1) $2 tan^{- 1} (\frac{1 + x + x ^{2}}{x}) + c$ (2) $tan^{- 1} (\frac{1 + x + x ^{2}}{x}) + c$ (3) $tan^{- 1} (\frac{x}{1 + x + x ^{2}}) + c$ (4) $tan^{- 1} (\frac{1 + x ^{2}}{x}) + c$ 200. $The value of the integral \int \frac{sin θ \cdot sin 2 θ ( sin ^{6} θ + sin ^{4} θ + sin ^{2} θ ) 2 sin ^{4} θ + 3 sin ^{2} θ + 6}{1 - cos 2 θ} dθ is (where c is a constant of integration$ (1) $\frac{1}{18} (9 - 2 sin^{6} θ - 3 sin^{4} θ - 6 sin^{2} θ)^{\frac{3}{2}} + c$ (2) $\frac{1}{18} (11 - 18 sin^{2} θ + 9 sin^{4} θ - 2 sin^{6} θ)^{\frac{3}{2}} + c$ (3) $\frac{1}{18} (11 - 18 cos^{2} θ + 9 cos^{4} θ - 2 cos^{6} θ)^{\frac{3}{2}} + c$ (4) $\frac{1}{18} (9 - 2 cos^{6} θ - 3 cos^{4} θ - 6 cos^{2} θ)^{\frac{3}{2}} + c$ 201. $For k \in (1, \infty), \int \frac{1}{1 + kcos x} dx =$ (1) $\frac{2}{1 + k ^{2}} tan^{- 1} (\frac{1 - k}{1 + k} tan \frac{x}{2}) + C$ (2) $\frac{1}{k ^{2} - 1} lo g (\frac{k + 1 + k - 1 t a n \frac{x}{2}}{k + 1 - k - 1 t a n \frac{x}{2}}) + C$ (3) $\frac{1}{k ^{2} + 1} lo g (\frac{k + 1 + k - 1 t a n \frac{x}{2}}{k + 1 - k - 1 t a n \frac{x}{2}}) + C$ (4) $\frac{1}{k ^{2} - 1} tan^{- 1} (\frac{k - 1 c o s \frac{x}{2} + k - 1 s i n \frac{x}{2}}{k + 1 c o s \frac{x}{2} - k - 1 s i n \frac{x}{2}}) + C$ 202. $Evaluate \int sin (k) dk on (0, \infty)$ (1) $2 [cos (k) - k sin (k)] + c$ (2) $2 [cos (k) + k sin (k)] + c$ (3) $2 [k cos (k) - k sin (k)] + c$ (4) $2 [sin (k) - k cos (k)] + c$ 203. $If \int e^{x} cos x d x = \frac{e ^{x}}{2} (cos x + sin x) and \int \frac{c o s ( l o g ( \frac{2 x + 3}{3 - 2 x} ) )}{( 3 - 2 x ) ^{2}} d x = \frac{f ( x )}{24} [cos (g (x)) + sin (g (x))] + c, then g (1) =$ (1) $5$ (2) $log f (2)$ (3) $log f (1)$ (4) $0$ 204. $\int_{- 5}^{5} \| x + 2 \| dx is equal to$ (1) $28$ (2) $29$ (3) $27$ (4) $30$ 205. $If \int \frac{sin ( x - \frac{π}{4} )}{2 + sin 2 x} dx = - \frac{1}{2} tan^{- 1} (f (x)) + C, then f (x) =$ (1) $sin x - cos x$ (2) $2 cos (x - \frac{π}{4})$ (3) $sin (x - \frac{π}{4})$ (4) $2 tan (x - \frac{π}{4})$ 206. $\int \frac{x ^{8} + 4}{x ^{4} - 2 x ^{2} + 2} dx = A x^{5} + B x^{3} + Cx + K, then 5 A + 3 B + C =$ (1) $7$ (2) $5$ (3) $3$ (4) $1$ 207. $\int_{- \frac{π}{4}}^{\frac{π}{4}} (\frac{x + \frac{π}{4}}{2 - cos 2 x}) dx is equal to$ (1) $\frac{8 π 3}{5}$ (2) $\frac{2 π 3}{9}$ (3) $\frac{4 π ^{2} 3}{9}$ (4) $\frac{π ^{2}}{6 3}$ 208. $\int_{0}^{π /4} [tan x + cot x] dx is equal to$ (1) $\frac{π}{2}$ (2) $\frac{π}{2}$ (3) $\frac{3 π}{2}$ (4) $π$ 209. $lim n \to \infty \frac{1}{n ^{3}} \sum (k^{2} x) =$ (1) $x$ (2) $\frac{x}{2}$ (3) $\frac{x}{3}$ (4) $\frac{x}{4}$ 210. $Let f (x) = \int \frac{d x}{x ^{2/3} + 2 x ^{1/2}} be such that f (0) = - 26 + 24 lo g_{e} (2) . If f (1) = a + b lo g_{e} (3), where a, b \in Z, then a + b is equal to :$ (1) $-26$ (2) $-11$ (3) $-5$ (4) $-18$