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JEE • MATHEMATICS • INDEFINITE INTEGRATION

Indefinite integration practice · Medium

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271 Medium
271 of 411
Application +4 | -1
If I=∫1/ee∣log⁡x∣dxx2I=\int_{1 / \mathrm{e}}^{\mathrm{e}}|\log \mathrm{x}| \frac{\mathrm{dx}}{\mathrm{x}^2}, then I equals
View answer and explanation
Correct option: C
2(1−1/e)2(1-1 / \mathrm{e})
Option C (2(1−1/e)2(1-1 / \mathrm{e})) is the correct answer because it is the result of evaluating the definite integral after splitting it into two parts based on the absolute value of the logarithm.
272 Medium
272 of 411
Application +4 | -1
∫(cos⁡xx−xcos⁡x⋅sin⁡x)dx\int\left(\sqrt{\frac{\cos x}{x}}-\sqrt{\frac{x}{\cos x}} \cdot \sin x\right) d x equals
View answer and explanation
Correct option: C
2xcos⁡x+C2 \sqrt{x \cos x}+C
Option C (2xcos⁡x+C2 \sqrt{x \cos x}+C) is correct because it is the result of the given integral after substitution.
273 Medium
273 of 411
Application +4 | -1
∫log⁡ex(1+log⁡ex)2dx=f(x)1+log⁡ex+c\int \frac{\log _e x}{\left(1+\log _e x\right)^2} d x=\frac{f(x)}{1+\log _e x}+c where f(x)f(x) is
View answer and explanation
Correct option: A
xx
Option A (xx) is correct because after substitution and integration, f(x) is identified as x.
274 Medium
274 of 411
Application +4 | -1
∫(∫ex(ln⁡x+2x−1x2)dx)dx=\int\left(\int e^x\left(\ln x+\frac{2}{x}-\frac{1}{x^2}\right) d x\right) d x=
View answer and explanation
Correct option: A
exln⁡x+C1x+C2e^x \ln x+C_1 x+C_2
Option A (exln⁡x+C1x+C2e^x \ln x+C_1 x+C_2) is correct because it results from correctly integrating by parts twice and simplifying the expression.
275 Medium
275 of 411
Analyzing +4 | -1
∫ex(x3+x+1)(x2+1)3/2dx=\int e^x \frac{\left(x^3+x+1\right)}{\left(x^2+1\right)^{3 / 2}} d x=
View answer and explanation
Correct option: B
xexx2+1+c\frac{x e^x}{\sqrt{x^2+1}}+c
Option B (GP) is correct because the sequence consisting of odd terms (1, 25, 625, ...) has a common ratio between consecutive terms, which defines a geometric progression.
276 Medium
276 of 411
Analyzing +4 | -1
∫  {log⁡  x−11+(log⁡  x)  2}2dx\int \,\,{\left\{ {\frac{{\log \,\,x - 1}}{{1 + (\log \,\,x){\;^2}}}} \right\}^2}dx is equal to
View answer and explanation
Correct option: A
x(log⁡  x)2+  1+  c\frac{x}{{{{(\log \,\,x)}^2} + \,\,1}} + \,\,c
Option A (x(log⁡ x)2+ 1+ c\frac{x}{{{{(\log \,x)}^2} + \,1}} + \,c) is correct because the derivative of x(log⁡ x)2+ 1\frac{x}{{{{(\log \,x)}^2} + \,1}} is equal to the given integrand.
277 Medium
277 of 411
Application +4 | -1
∫sin⁡xsin⁡x−cos⁡xdx\int \frac{\sin x}{\sin x-\cos x} d x is equal to
View answer and explanation
Correct option: B
x2+12log⁡(sin⁡x−cos⁡x)+c\frac{x}{2}+\frac{1}{2} \log (\sin x-\cos x)+c
Option B (x2+12log⁡(sin⁡x−cos⁡x)+c\frac{{x}}{{2}}+\frac{{1}}{{2}} \log (\sin x-\cos x)+c) is correct because it is the result of correctly applying the method of substitution and integration.
278 Medium
278 of 411
Application +4 | -1
∫tan⁡ (sin⁡−1 x)   dx\int \tan \,({\sin ^{ - 1}}\,x)\,\;dx is equal to
View answer and explanation
Correct option: D
− 1  −  x2+  c- \,\sqrt {1\,\, - \,\,{x^2}} + \,\,c
Option D (\ - \,\sqrt {1\\,\\ - \\,{x^2}} + \\,\c\ - \,\sqrt {1\\,\\ - \\,{x^2}} + \\,\c) is correct because the integral of tan⁡(sin⁡−1x)\tan(\sin^{-1}x) can be solved by trigonometric substitution and simplification.
279 Medium
279 of 411
Application +4 | -1
∫dx5+4cos⁡x\int \frac{d x}{5+4 \cos x} is equal to
View answer and explanation
Correct option: A
23tan⁡−1(13tan⁡x2)+c\frac{2}{3} \tan ^{-1}\left(\frac{1}{3} \tan \frac{x}{2}\right)+c
Option A (23tan⁡−1(13tan⁡x2)+c\frac{2}{3} \tan ^{-1}(\frac{1}{3} \tan \frac{x}{2})+c) is correct because it is the result of integrating the given expression using trigonometric identities and substitution.
280 Medium
280 of 411
Application +4 | -1
∫tan⁡xsec⁡2x1−tan⁡2xdx=\int \tan x \sec ^2 x \sqrt{1-\tan ^2 x} d x=
View answer and explanation
Correct option: A
−13(1−tan⁡2x)3/2+c-\frac{1}{3}\left(1-\tan ^2 x\right)^{3 / 2}+c
Option A (−13(1−tan⁡2x)3/2+c-\frac{1}{3}\left(1-\tan ^2 x\right)^{3 / 2}+c) is correct because the integral simplifies to this form after substitution.
281 Medium
281 of 411
Application +4 | -1
∫3x2x6+1dx=\int \frac{3 x^2}{x^6+1} d x=
View answer and explanation
Correct option: B
tan⁡−1(x3)+c\tan ^{-1}\left(x^3\right)+c
Option B is correct because the integral simplifies to tan⁡−1(x3)+c\tan ^{-1}(x^3)+c after a simple substitution.
282 Medium
282 of 411
Application +4 | -1
∫(tan⁡  x  +  cot⁡  x)dx\int \left( {\sqrt {\tan \,\,x} \,\, + \,\,\sqrt {\cot \,\,x} } \right)dx is equal to
View answer and explanation
Correct option: A
2  sin⁡−1 (sin⁡  x  −  cos⁡  x)  +  c\sqrt 2 \,\,{\sin ^{ - 1}}\,(\sin \,\,x\,\, - \,\,\cos \,\,x)\,\, + \,\,c
Option A is correct because the integral simplifies to a form that can be solved using a trigonometric substitution, resulting in the given arcsine function.
283 Medium
283 of 411
Application +4 | -1
The value of ∫x⋅ℓn(x+1+x2)1+x2dx\int x \cdot \frac{\ell n \left(x+\sqrt{1+x^2}\right)}{\sqrt{1+x^2}} d x equals:
View answer and explanation
Correct option: A
1+x2ℓn(x+1+x2)−x+C\sqrt{1+x^2} \ell n \left(x+\sqrt{1+x^2}\right)-x+C
Option A (1+x2ℓn(x+1+x2)−x+C\sqrt{1+x^2} \ell n \left(x+\sqrt{1+x^2}\right)-x+C) is correct because it is the result of integration by parts applied to the given integral.
284 Medium
284 of 411
Application +4 | -1
∫01  cot⁡−1  (1−x+x2)    dx\int \nolimits_0^1 \,\,{\cot ^{ - 1}}\,\,\left( {1 - x + {x^2}} \right)\;\,\,dx is equal to
View answer and explanation
Correct option: D
π2  −  log⁡  2\frac{\pi }{2}\,\, - \,\,\log \,\,2
Option D (π2 − log⁡ 2\frac{\pi }{2}\, - \,\log \,2) is correct because after rewriting the integral and using integration by parts, the result is π2−log⁡2\frac{\pi}{2} - \log 2.
285 Medium
285 of 411
Application +4 | -1
If ∫3sin⁡x+2cos⁡x2sin⁡x+3cos⁡xdx=lx+mlog⁡e∣2sin⁡x+3cos⁡x∣+c\int \frac{3 \sin x+2 \cos x}{2 \sin x+3 \cos x} d x=l x+m \log _e|2 \sin x+3 \cos x|+c then
View answer and explanation
Correct option: B
l=1213,m=−513l=\frac{12}{13}, m=-\frac{5}{13}
Option B (l=1213,m=−513l=\frac{{12}}{{13}}, m=-\frac{{5}}{{13}}) is correct because it is obtained by applying the formula derived from decomposing the integrand.

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SEQUENCE OF SERIES 1764 MCQs DEFINITE INTEGRATION 1443 MCQs MATRICES 1363 MCQs PROBABILITY 1146 MCQs COMPLEX NUMBERS 980 MCQs DIFFERENTIAL EQUATIONS 977 MCQs
286 Medium
286 of 411
Application +4 | -1
If ∫(sin⁡2x−cos⁡2x)dx=12sin⁡(2x−a)+b\int(\sin 2 x-\cos 2 x) d x=\frac{1}{\sqrt{2}} \sin (2 x-a)+b, then
View answer and explanation
Correct option: D
a=−5π4,b=a=-\frac{5 \pi}{4}, b= any constant
Option B ([1,3/2]) is correct because it represents the interval where the expression inside the square root is non-negative.
287 Medium
287 of 411
Application +4 | -1
∫dxtan⁡x+cot⁡x+sec⁡x+cosec⁡x\int \frac{d x}{\tan x+\cot x+\sec x+\operatorname{cosec} x} is equal to
View answer and explanation
Correct option: C
12(sin⁡x−cos⁡x−x)+c\frac{1}{2}(\sin x-\cos x-x)+c
Option C (12(sin⁡x−cos⁡x−x)+c\frac{1}{2}(\sin x-\cos x-x)+c) is correct because after simplifying the integrand and integrating, this is the resulting expression.
288 Medium
288 of 411
Application +4 | -1
∫sec⁡32θdθ\int \sec ^3 2 \theta d \theta is equal to
View answer and explanation
Correct option: B
14sec⁡2θtan⁡2θ+ln⁡sec⁡2θ+tan⁡2θ+c\frac{1}{4} \sec 2 \theta \tan 2 \theta+\ln \sqrt{\sec 2 \theta+\tan 2 \theta}+c
Option B (14sec⁡2θtan⁡2θ+ln⁡sec⁡2θ+tan⁡2θ+c\frac{1}{4} \sec 2 \theta \tan 2 \theta+\ln \sqrt{\sec 2 \theta+\tan 2 \theta}+c) is correct because it is the result of integrating sec⁡32θ\sec^3 2\theta using integration by parts and standard integral formulas.
289 Medium
289 of 411
Application +4 | -1
To evaluate ∫x3e3x2+5dx\int x^3 e^{3 x^2+5} d x, the simplest way is to
View answer and explanation
Correct option: B
Substitute (3x2+5)=t\left(3 x^2+5\right)=t
Option B (Substitute (3x2+5)=t\left(3 x^2+5\right)=t) is correct because this substitution simplifies the integral into a manageable form.
290 Medium
290 of 411
Application +4 | -1
∫  (2x12+  5x9)(x5+  x3+  1)3    dx\int \,\,\frac{{\left( {2{x^{12}} + \,\,5{x^9}} \right)}}{{{{\left( {{x^5} + \,\,{x^3} + \,\,1} \right)}^3}}}\,\,\;dx is equal to
View answer and explanation
Correct option: B
x102  (x5+  x3  +1)2  +  c\frac{{{x^{10}}}}{{2\,\,{{\left( {{x^5} + \,\,{x^3}\,\, + 1} \right)}^2}}}\,\, + \,\,c
Option B (x102  (x5+ x3 +1)2 + c\frac{{{x^{10}}}}{{2\,\,{{\left( {{x^5} + \,{x^3}\, + 1} \right)}^2}}}\, + \,c\) is correct because after substituting and integrating, we obtain the given result.
291 Medium
291 of 411
Application +4 | -1
For what value of aa and bb, the equation ∫(sin⁡2x−cos⁡2x)dx=12sin⁡(2x−a)+b holds good.\int(\sin 2 x-\cos 2 x) d x=\frac{1}{\sqrt{2}} \sin (2 x-a)+b \text { holds good.}
View answer and explanation
Correct option: A
a=−5π4,ba=-\frac{5 \pi}{4}, b is any arbitrary constant
Option A (a=−5π4a = -\frac{{5 \pi}}{{4}}, b is any arbitrary constant) is correct because integrating and comparing with the given form yields the value of a.
292 Medium
292 of 411
Analyzing +4 | -1
Let g(x)g(x) be a differentiable function satisfying ddx{g(x)}=g(x)\frac{d}{d x}\{g(x)\}=g(x) and g(0)=1g(0)=1, then
∫g(x)(2−sin⁡2x1−cos⁡2x)dx\int g(x)\left(\frac{2-\sin 2 x}{1-\cos 2 x}\right) d x is equal to
View answer and explanation
Correct option: B
−g(x)cot⁡x+C-g(x) \cot x+C
The correct answer is B (−g(x)cot⁡x+C-g(x) \cot x+C) because given the properties of g(x)g(x), the integral simplifies and can be solved using integration techniques.
293 Medium
293 of 411
Application +4 | -1
∫x2(9−x2)3/2dx=\int \frac{x^2}{\left(9-x^2\right)^{3 / 2}} d x=
View answer and explanation
Correct option: A
x9−x2−sin⁡−1x3+c\frac{x}{\sqrt{9-x^2}}-\sin ^{-1} \frac{x}{3}+c
Option A is correct because the integral can be solved using trigonometric substitution and simplifying the resulting expression.
294 Medium
294 of 411
Application +4 | -1
∫(sin⁡x+cos⁡x)(2−sin⁡2x)sin⁡22xdx=\int \frac{(\sin x+\cos x)(2-\sin 2 x)}{\sin ^2 2 x} d x=
View answer and explanation
Correct option: B
sin⁡x−cos⁡xsin⁡2x+c\frac{\sin x-\cos x}{\sin 2 x}+c
Option B (sin⁡x−cos⁡xsin⁡2x+c\frac{\sin x-\cos x}{\sin 2 x}+c) is correct because the integral simplifies to this expression after a suitable substitution.
295 Medium
295 of 411
Application +4 | -1
∫  exlog⁡a ex  dx\int \,\,{e^{x\log a}}\,{e^x}\;dx is equal to
View answer and explanation
Correct option: D
(ae)xlog⁡e  ae  +  c\frac{{{{\left( {ae} \right)}^x}}}{{{{\log }_e}\,\,ae}}\,\, + \,\,c
Option D ((ae)xlog⁡e ae + c\frac{{{{\left( {ae} \right)}^x}}}{{{{\log }_e}\,ae}}\, + \,c\) is correct because the integral of exlog⁡aexe^{x\log a}e^x simplifies to (ae)xlog⁡e ae + c\frac{{{{\left( {ae} \right)}^x}}}{{{{\log }_e}\,ae}}\, + \,c.
296 Medium
296 of 411
Application +4 | -1
If ∫x+sin⁡x1+cos⁡x dx=f(x)tan⁡x2+C\int \frac{x+\sin x}{1+\cos x} \mathrm{~d} x=f(x) \tan \frac{x}{2}+C
View answer and explanation
Correct option: B
f(x)=xf(x)=x
Option B (f(x)=xf(x)=x) is correct because after integration and simplification, the given integral equals xtan⁡x2+Cx \tan \frac{x}{2}+C.
297 Medium
297 of 411
Application +4 | -1
The value of ∫xdx1+x2+(1+x2)3\int \frac{x d x}{\sqrt{1+x^2+\sqrt{\left(1+x^2\right)^3}}} is equal to:
View answer and explanation
Correct option: B
21+1+x2+C2 \sqrt{1+\sqrt{1+x^2}}+C
Option B (21+1+x2+C2 \sqrt{1+\sqrt{1+x^2}}+C) is correct because it is the result of a u-substitution that simplifies the integral.
298 Medium
298 of 411
Application +4 | -1
The value of ∫dxx(xn+1)\int {\frac{{dx}}{{x\left( {{x^n} + 1} \right)}}} is equal to
View answer and explanation
Correct option: D
1nlog⁡∣xnxn+1∣+c\frac{1}{n}\log \left| {\frac{{{x^n}}}{{{x^n} + 1}}} \right| + c
Option D (1nlog⁡∣xnxn+1∣+c\frac{1}{n}\log \left| {\frac{{{x^n}}}{{{x^n} + 1}}} \right| + c) is correct because by substituting t=xnt = x^n, the integral simplifies and can be solved using partial fractions.
299 Medium
299 of 411
Application +4 | -1
∫dx(2x+1)(1+(2x+1))\int \frac{d x}{(2 x+1)(1+\sqrt{(2 x+1)})} is equal to
View answer and explanation
Correct option: B
log⁡e2x+11+2x+1+c\log _e \frac{\sqrt{2 x+1}}{1+\sqrt{2 x+1}}+c
Option B (log⁡e2x+11+2x+1+c\log _e \frac{\sqrt{2 x+1}}{1+\sqrt{2 x+1}}+c) is correct because the integral is solved using a substitution and partial fraction decomposition.
300 Medium
300 of 411
Analyzing +4 | -1
∫xm−1xm+11−2xm+mx2mdx=f(x)mxm+c\int \frac{x^m-1}{x^{m+1} \sqrt{1-2 x^m+m x^{2 m}}} d x=\frac{f(x)}{m x^m}+c where f(x)f(x) is equal to
View answer and explanation
Correct option: A
1−2xm+mx2m\sqrt{1-2 x^m+m x^{2 m}}
Option A (1−2xm+mx2m\sqrt{{1-2 x^m+m x^{2 m}}}) is correct because the integral simplifies to this expression after substitution and algebraic manipulation.
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