Indefinite Integration PYQs Set 6 (PDF)

Set 6 of 16 · Questions 151-180 of 459

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Indefinite Integration · Set 6 (Q151-180)
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151. $\int \frac{y ^{2} + 3 y ^{4} + 6 y ^{2}}{y ( 1 + 3 y ^{2} )} d y =$ (1) $\frac{3}{4} 3 y^{4} + 3 tan^{- 1} (3 y) + C$ (2) $\frac{3}{2} y^{2/3} + 6 tan^{- 1} (6 y^{2}) + C$ (3) $\frac{2}{3} 3 y^{2} + 6 lo g (1 + y^{2}) + C$ (4) $3 (1 + y) + tan^{- 1} (y^{3/2}) + C$ 152. $If \frac{13 x + 43}{2 x ^{2} + 17 x + 30} = \frac{A}{2 x + 5} + \frac{B}{x + 6} then A^{2} + B^{2} =$ (1) $22/3$ (2) $52$ (3) $34$ (4) $18/5$ 153. $If (\int \frac{1 - ( cot x ) ^{2019}}{tan x + ( cot x ) ^{2020}} dx = \frac{1}{n} ln) (f (x))^{n} + (g (x))^{n} ∣ + c, then the value of n ((f (x))^{4} + (g (x))^{4})_{x = \frac{π}{3}} =$ (1) $\frac{10105}{16}$ (2) $\frac{10012}{15}$ (3) $\frac{20210}{9}$ (4) $\frac{10105}{8}$ 154. $If n \in N and I_{n} = \int (log x)^{n} dx, then I_{n} + n I_{n - 1} =$ (1) $\frac{( log x ) ^{n}}{n}$ (2) $\frac{( log x ) ^{n + 1}}{n + 1}$ (3) $x (log x)^{n}$ (4) $(log x)^{n - 1}$ 155. $\sum \frac{x ^{e - 1} + e ^{x - 1}}{x ^{e} + e ^{x}} dx =$ (1) $\frac{- 1}{e} log (x^{e} + e^{x}) + C$ (2) $- elog (x^{e} + e^{x}) + C$ (3) $\frac{1}{e} log (x^{e} + e^{x}) + C$ (4) $elog (x^{e} + e^{x}) + C$ 156. $If linear function f (x) and g (x) satisfy \int [(3 x - 1) cos x + (1 - 2 x) sin x] dx = f (x) cos x + g (x) sin x + C, then$ (1) $f (x) = 3 (x - 1)$ (2) $f (x) = 3 x - 5$ (3) $g (x) = 3 (x - 1)$ (4) $g (x) = 3 + x$ 157. $The positive integer n \leq 5 for which \int_{0}^{1} e^{x} (x - 1)^{n} dx = 16 - 6 e is$ (1) $5$ (2) $4$ (3) $3$ (4) $2$ 158. $If \frac{2 x ^{2} - 3 x + 5}{( x - 7 ) ^{3}} = \frac{A}{x - 7} + \frac{B}{( x - 7 ) ^{2}} + \frac{C}{( x - 7 ) ^{3}} then 2 A - 3 B + C =$ (1) $0$ (2) $27$ (3) $11$ (4) $15$ 159. $If I_{n} = \int tan^{n} xdx, and I_{0} + I_{1} + 2 I_{2} + 2 I_{3} + 2 I_{4} + I_{5} + I_{6} = \sum_{K = 1}^{n} \frac{tan ^{K} x}{K}, then n =$ (1) $6$ (2) $5$ (3) $4$ (4) $3$ 160. $\int \frac{1}{1 + 3 sin ^{2} x + 8 cos ^{2} x} dx =$ (1) $\frac{1}{6} tan^{- 1} (\frac{2 tan x}{3}) + C$ (2) $\frac{1}{6} tan^{- 1} (2 tan x) + C$ (3) $6 tan^{- 1} (\frac{2 tan x}{3}) + C$ (4) $tan^{- 1} (\frac{2 tan x}{3}) + C$ 161. $The value of \int \frac{( x ^{2} + 1 )}{( x ^{4} - x ^{2} + 1 ) cot ^{- 1} ( x - \frac{1}{x} )} dx will be$ (1) $ln cot^{- 1} (x - \frac{1}{x}) + c$ (2) $- ln cot^{- 1} (x - \frac{1}{x}) + c$ (3) $ln cot^{- 1} (x + \frac{1}{x}) + c$ (4) $None of these$ 162. $If \int e^{sin^{2} x} (sin xcos x + cos^{3} xsin x) dx = e^{sin^{2} x} (1 + f (x)) + c then f^{’} (x) =$ (1) $\frac{1}{2} sin^{2} x$ (2) $\frac{1}{2} cos^{2} x$ (3) $- \frac{1}{2} cos 2 x$ (4) $- \frac{1}{2} sin 2 x$ 163. $If f (x) = \int \frac{16 x ^{7} + 5 x ^{10}}{( x ^{3} + 2 + 3 x ^{8} ) ^{2}} dx (x \geq 0) and f (0) = 1, then the value of f (- 1) is$ (1) $\frac{7}{6}$ (2) $\frac{5}{4}$ (3) $\frac{- 3}{4}$ (4) $\frac{- 5}{6}$ 164. $If A (x) = x + 1 2 x + 1 3 x + 1 2 x + 1 3 x + 1 x + 1 3 x + 1 x + 1 2 x + 1, then \int_{0}^{1} A (x) d x is equal to$ (1) $-15 .$ (2) $- \frac{15}{2}$ (3) $-30$ (4) $-5$ 165. $If \frac{- x ^{2} + 6 x + 1}{( x - 1 ) ^{2} ( x ^{2} + 2 )} = \frac{A}{x - 1} + \frac{B}{( x - 1 ) ^{2}} + \frac{Cx - 3}{x ^{2} + 2} A + B + C =$ (1) $7$ (2) $5$ (3) $3$ (4) $2$ 166. $The value of \int \frac{x ^{2} dx}{x ^{6} + a ^{6}} is equal to$ (1) $log (x^{3} + x^{6} + a^{6}) + C$ (2) $log (x^{3} - x^{6} + a^{6}) + C$ (3) $\frac{1}{3} log (x^{3} + x^{6} + a^{6}) + C$ (4) $\frac{1}{3} log (x^{3} - x^{6} + a^{6}) + C$ 167. $\int \frac{dx}{cot ^{2} x - 1} = \frac{1}{A} log \| sec 2 x + tan 2 x \| - \frac{x}{B} + c, (where c is constant of integration), then A + B =$ (1) $-6$ (2) $6$ (3) $-5$ (4) $5$ 168. $If \int \frac{cos θ}{5 + 7 sin θ - 2 cos ^{2} θ} dθ = Alog_{e} \| B (θ) \| + C where C is a constant of integration, then \frac{B ( θ )}{A} can be:$ (1) $\frac{2 sin θ + 1}{5 ( sin θ + 3 )}$ (2) $\frac{2 sin θ + 1}{sin θ + 3}$ (3) $\frac{5 ( 2 sin θ + 1 )}{sin θ + 3}$ (4) $\frac{5 ( sin θ + 3 )}{2 sin θ + 1}$ 169. $\int_{- 1}^{1} \frac{log ( 1 + x )}{1 + x ^{2}} dx = \int_{0}^{1} \frac{log ( 1 + x )}{1 + x ^{2}} dx + \int_{0}^{1} f (x) dx, then f (x) is equal to$ (1) $\frac{log ( 1 + x )}{1 + x ^{2}}$ (2) $- \frac{log ( 1 + x )}{1 + x ^{2}}$ (3) $\frac{log ( 1 - x )}{1 + x ^{2}}$ (4) $0$ 170. $If \int (cos (x) \cdot \frac{d}{dx} (cosec (x))) dx = f (x) + g (x) + c, then f (x) \cdot g (x) is equal to$ (1) $xcot (x)$ (2) $xtan (x)$ (3) $xcos (x)$ (4) $1$ 171. $\int \frac{sec x}{sec x + tan x} dx =$ (1) $sec x + tan x + C$ (2) $log sin x + log cos x + C$ (3) $tan x - sec x + C$ (4) $log (1 + sin x) + C$ 172. $\sum \frac{sin x \cdot sec ^{2} x - tan x \cdot sin x + cos x}{( 1 - cos 2 x )} dx =$ (1) $\frac{1}{2} (sec x - cosec x - log (tan (\frac{x}{2}) tan (\frac{π}{4} + \frac{x}{2}))) + c$ (2) $sec x + cosec x + log (\frac{tan ( \frac{x}{2} )}{tan ( \frac{π}{4} + \frac{x}{2} )}) + c$ (3) $\frac{1}{2} (sec x - cosec x - log (\frac{tan ( \frac{π}{4} + \frac{x}{2} )}{tan ( \frac{x}{2} )})) + c$ (4) $sec x + cosec x - log (tan (\frac{x}{2})) + c$ 173. $If \frac{3 x ^{4} + 5 x ^{2} + 2}{( x ^{2} + 1 ) ^{2} ( x ^{2} + 2 )} = \frac{Ax + B}{x ^{2} + 2} + \frac{Cx + D}{x ^{2} + 1} + \frac{Ex + F}{( x ^{2} + 1 ) ^{2}} then A + 2 B + D + 4 E =$ (1) $5$ (2) $0$ (3) $1$ (4) $7$ 174. $\int_{0}^{π /2} \frac{cos x sin x}{1 + sin x} dx is equal to$ (1) $log 2 - 1$ (2) $log 2$ (3) $- log 2$ (4) $1 - log 2$ 175. $\int_{- π /2}^{π /2} \frac{dx}{e ^{sin x} + 1} is equal to$ (1) $1$ (2) $0$ (3) $\frac{π}{2}$ (4) $- \frac{π}{2}$ 176. $If I = \int_{1}^{3} 3 + x + x^{2} dx, then I lies in the interval$ (1) $(25, 215)$ (2) $(3, 25)$ (3) $(23, 33)$ (4) $(215, 23)$ 177. $\int \frac{x ^{2} + 1}{x ^{4} + 7 x ^{2} + 1} dx is equal to$ (1) $\frac{1}{3} tan^{- 1} (\frac{x ^{2} - 1}{3 x}) + C$ (2) $tan^{- 1} (\frac{x ^{2} - 1}{x}) + C$ (3) $\frac{1}{3} tan^{- 1} (\frac{x ^{2} - 1}{x}) + C$ (4) $\frac{1}{3} tan^{- 1} (\frac{x ^{2} - 1}{3 x}) + C$ 178. $If \int e^{2 x} f^{’} (x) dx = g (x), then \int (e^{2 x} f (x) + e^{2 x} f^{’} (x)) dx =$ (1) $\frac{1}{2} (e^{2 x} f (x) - g (x)) + C$ (2) $\frac{1}{2} (e^{2 x} f (x) + g (x)) + C$ (3) $\frac{1}{2} (e^{2 x} f (2 x) + g (x)) + C$ (4) $\frac{1}{2} (e^{2 x} f^{’} (x) + g (x)) + C$ 179. $\sum \frac{cos 2 x}{sin x} dx =$ (1) $\frac{1}{2 2} log (\frac{2 + 1 - tan ^{2} x}{2 - 1 - tan ^{2} x}) - \frac{1}{2} log (\frac{1 - 1 - tan ^{2} x}{1 + 1 - tan ^{2} x}) + c$ (2) $\frac{1}{2} log (\frac{2 + 1 - tan ^{2} x}{2 - 1 - tan ^{2} x}) - \frac{1}{2} log (\frac{1 + 1 - tan ^{2} x}{1 - 1 - tan ^{2} x}) + c$ (3) $\frac{1}{4 2} log (\frac{2 - 1 - tan ^{2} x}{2 + 1 - tan ^{2} x}) + \frac{1}{2} log (\frac{1 - 1 - tan ^{2} x}{1 + 1 - tan ^{2} x}) + c$ (4) $(\frac{1}{4 2} log) (\frac{2 - 1 - tan ^{2} x}{2 + 1 - tan ^{2} x}) + \frac{1}{2 2} log (\frac{1 - 1 - tan ^{2} x}{1 + 1 - tan ^{2} x}) + c$ 180. $If \int (\frac{1}{x} + \frac{1}{x ^{3}}) 23 3 x^{- 24} + x^{- 26} d x = - \frac{α}{3 ( α + 1 )} (3 x^{β} + x^{γ})^{\frac{α + 1}{α}} + C, x > 0, (α, β, γ \in Z), where C is the constant of integration, then α + β + γ is equal to ________.$