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JEE • MATHEMATICS • INDEFINITE INTEGRATION

Indefinite integration practice · Mixed difficulty

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421 Easy
421 of 995
Application +4 | -1
∫−π/4π/4dx1+cos⁡2x\int\limits_{{-\pi / 4}}^{\pi / 4} \frac{d x}{1+\cos 2 x} is equal to
View answer and explanation
Correct option: A
1
Option A (1) is correct because the integral simplifies to the integral of sec²(x) from 0 to π/4, which equals tan(π/4) - tan(0) = 1.
422 Easy
422 of 995
Application +4 | -1
If ∫(1−tan⁡3x)2dx=13[tan⁡3x+log⁡f(x)]+C\int(1-\tan 3 x)^2 d x=\frac{1}{3}[\tan 3 x+\log f(x)]+C then f(x)f(x) is given by
View answer and explanation
Correct option: B
cos⁡23x\cos ^2 3 x
Option B (cos⁡23x\cos ^2 3 x) is correct because after expanding and integrating, the result matches the given form with f(x)=cos⁡23xf(x) = \cos^2 3x.
423 Medium
423 of 995
Application +4 | -1
∫e2x−1e2x+1dx=\int \frac{e^{2 x}-1}{e^{2 x}+1} d x=
View answer and explanation
Correct option: B
log⁡(e2x+1)−x+c\log \left(e^{2 x}+1\right)-x+c
Option B is correct because the integral can be solved by rewriting the integrand and using substitution.
424 Easy
424 of 995
Application +4 | -1
If the antiderivative of 1x+x3/2\frac{1}{\sqrt{x+x^{3 / 2}}} is A1+x+CA \sqrt{1+\sqrt{x}}+C then AA is equal to
View answer and explanation
Correct option: D
4
Option D (4) is correct. By substituting \x1/2=tx^{1/2} = t, the integral simplifies, leading to the antiderivative \41+x+C4 \sqrt{1+\sqrt{x}} + C, thus A = 4.
425 Easy
425 of 995
Analyzing +4 | -1
If ∫sin⁡8x−cos⁡8x1−2sin⁡2xcos⁡2xdx=asin⁡2x+C\int \frac{\sin ^8 x-\cos ^8 x}{1-2 \sin ^2 x \cos ^2 x} d x=a \sin 2 x+C, then a=a=
View answer and explanation
Correct option: A
−12-\frac{1}{2}
Option A is correct because after simplifying the integral, we get −12sin⁡2x+C-\frac{1}{2} \sin 2x + C, which means a=−12a = -\frac{1}{2}.
426 Medium
426 of 995
Analyzing +4 | -1
∫ex2(2x+x3)(3+x2)2dx\int \frac{e^{x^2}\left(2 x+x^3\right)}{\left(3+x^2\right)^2} d x is equal to:
View answer and explanation
Correct option: D
12ex2(3+x2)+c\frac{1}{2} \frac{e^{x^2}}{\left(3+x^2\right)}+c
The correct answer is D (12ex2(3+x2)+c\frac{1}{2} \frac{e^{x^2}}{\left(3+x^2\right)}+c) because the integral can be solved by recognizing a pattern related to the derivative of ex2e^{x^2} and applying appropriate simplification.
427 Easy
427 of 995
Application +4 | -1
∫1sin⁡x−cos⁡x+2dx\int \frac{1}{\sin x-\cos x+\sqrt{2}} d x equals
View answer and explanation
Correct option: D
−12cot⁡(x2+π8)-\frac{1}{\sqrt{2}} \cot \left(\frac{x}{2}+\frac{\pi}{8}\right)
Option D (−12cot⁡(x2+π8)-\frac{1}{\sqrt{2}} \cot \left(\frac{x}{2}+\frac{\pi}{8}\right)) is correct because it is the result of integrating the given expression after trigonometric simplification.
428 Easy
428 of 995
Analyzing +4 | -1
∫  x2−1x4+x2+1  dx\int \,\,\frac{{{x^2} - 1}}{{{x^4} + {x^2} + 1}}\,\,dx is equal to
View answer and explanation
Correct option: C
12  log⁡  x2  −  x  +  1x2  +  x  +  1  +  c\frac{1}{2}\,\,\log \,\,\frac{{{x^2}\,\, - \,\,x\,\, + \,\,1}}{{{x^2}\,\, + \,\,x\,\, + \,\,1}}\,\, + \,\,c
Option C (12  log⁡  x2 − x + 1x2 + x + 1 + c\frac{1}{2}\,\,\log \,\,\frac{{{x^2}\, - \,x\, + \,1}}{{{x^2}\, + \,x\, + \,1}}\, + \,c) is correct because after dividing by x2x^2 and using substitution, the integral can be solved to obtain this result.
429 Easy
429 of 995
Application +4 | -1
The value of ∫x2−1(x2+1)x4+1dx\int \frac{x^2-1}{\left(x^2+1\right) \sqrt{x^4+1}} d x, is equal to
View answer and explanation
Correct option: A
12sec⁡−1(x2+12x)+C\frac{1}{\sqrt{2}} \sec ^{-1}\left(\frac{x^2+1}{\sqrt{2} x}\right)+C
Option A (12sec⁡−1(x2+12x)+C\frac{{1}}{{\sqrt{{2}}}} \sec ^{-1}(\frac{{x^2+1}}{{\sqrt{{2}} x}})+C) is correct because the integral simplifies to this expression after dividing by x2x^2 and using a suitable substitution.
430 Easy
430 of 995
Application +4 | -1
∫  cos⁡  x−  sin⁡  x1+2  sin⁡  x  cos⁡  x  dx\int \,\,\frac{{\cos \,\,x - \,\,\sin \,\,x}}{{1 + 2\,\,\sin \,\,x\,\,\cos \,\,x}}\,\,dx is equal to
View answer and explanation
Correct option: C
−  1sin⁡  x  +  cos⁡  x  +  c- \,\,\frac{1}{{\sin \,\,x\,\, + \,\,\cos \,\,x}}\,\, + \,\,c
Option C ( −  1sin⁡ x +  cos⁡ x + c\ - \,\,\frac{1}{{\sin \,x\, + \,\,\cos \,x}}\, + \,c) is correct because it is the result of a u-substitution where u=sin⁡x+cos⁡xu = \sin x + \cos x.
431 Easy
431 of 995
Application +4 | -1
∫x2−1x32x4−2x2+1dx\int \frac{x^2-1}{x^3 \sqrt{2 x^4-2 x^2+1}} d x is equal to
View answer and explanation
Correct option: D
2x4−2x2+12x2+C\frac{\sqrt{2 x^4-2 x^2+1}}{2 x^2}+C
Option D is correct because after substitution and simplification, the integral can be solved, resulting in 2x4−2x2+12x2+C\frac{\sqrt{2 x^4-2 x^2+1}}{2 x^2}+C.
432 Medium
432 of 995
Application +4 | -1
∫x−3(sin⁡−1(ln⁡x)+cos⁡−1(ln⁡x))dx\int \sqrt{x-3}\left(\sin ^{-1}(\ln x)+\cos ^{-1}(\ln x)\right) d x is equal to
View answer and explanation
Correct option: D
None of these
Option D (None of these) is correct because the integrand is not defined for any real number, making the integral non-existent.
433 Easy
433 of 995
Application +4 | -1
The integral ∫(1+x−1x)ex+1xdx\int\left(1+x-\frac{1}{x}\right) e^{x+\frac{1}{x}} d x is equal to
View answer and explanation
Correct option: D
xex+1x+Cx e^{x+\frac{1}{x}}+C
Option D (xex+1x+Cx e^{x+\frac{1}{x}}+C) is correct because integrating by parts simplifies the expression to the given result.
434 Medium
434 of 995
Analyzing +4 | -1
∫dxsin⁡x+sec⁡x\int \frac{d x}{\sin x+\sec x} is equal to 123log⁡e∣3+t3−t∣+Tan⁡−1z+c\frac{1}{2 \sqrt{3}} \log _e\left|\frac{\sqrt{3}+t}{\sqrt{3}-t}\right|+\operatorname{Tan}^{-1} z+c where
View answer and explanation
Correct option: B
t=sin⁡x−cos⁡x,z=sin⁡x+cos⁡xt=\sin x-\cos x, z=\sin x+\cos x
Option A (t=sin⁡x−cos⁡x,z=sin⁡x+cos⁡xt=\sin x-\cos x, z=\sin x+\cos x) is correct because the given substitutions correctly separate the integral into two standard forms.
435 Easy
435 of 995
Application +4 | -1
∫xsec⁡2xdx=\int x \sec ^2 x d x=
View answer and explanation
Correct option: D
xtan⁡x+log⁡cos⁡x+cx \tan x+\log \cos x+c
Option D (xtan⁡x+log⁡cos⁡x+cx \tan x+\log \cos x+c) is correct because it is the result of integrating xsec⁡2xx \sec ^2 x using integration by parts.

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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
436 Easy
436 of 995
Application +4 | -1
If ∫f(x)dx=2{f(x)}3+C\int f(x) d x=2\{f(x)\}^3+C, then f(x)f(x) is
View answer and explanation
Correct option: D
x3\sqrt{\frac{x}{3}}
Option D is correct. Differentiating both sides of the given equation and solving the resulting differential equation yields the solution.
437 Medium
437 of 995
Understanding +4 | -1
∫f′  (x)f(x)  log⁡  {f(x)}dx=\int \frac{{f'\,\,\left( x \right)}}{{f\left( x \right)\,\,\log \,\,\left\{ {f\left( x \right)} \right\}}}dx =
View answer and explanation
Correct option: C
log⁡  {log⁡  f(x) }  +  C\log \,\,\left\{ {\log \,\,f\left( x \right)\,} \right\}\,\, + \,\,C
Option C (\log \,\left\{ {\log \,f\left( x \right)\} \right\}\, + \,C\log \,\left\{ {\log \,f\left( x \right)\} \right\}\, + \,C) is correct because it is the result of applying u-substitution twice.
438 Easy
438 of 995
Application +4 | -1
∫cos⁡−3/7x  sin⁡−11/7x dx\int {\cos ^{ - 3/7}}x\,\,{\sin ^{ - 11/7}}x\,dx is equal to
View answer and explanation
Correct option: C
−  74  tan⁡−4/7x+c- \,\,\frac{7}{4}\,\,{\tan ^{ - 4/7}}x + c
Option C (− 74 tan⁡−4/7x+c- \,\frac{7}{4}\,{{\tan }^{ - 4/7}}x + c) is correct because it is the result of integrating the given expression after simplification and substitution.
439 Medium
439 of 995
Understanding +4 | -1
Assertion: It is not possible to find ∫e−x2dx\int \mathrm{e}^{-\mathrm{x}^2} \mathrm{dx} by inspection method.

Reason : Function is not expressible in terms of elementary functions.
View answer and explanation
Correct option: A
Assertion is correct, Reason is correct; Reason is a correct explanation for assertion.
Assertion is correct, Reason is correct; Reason is a correct explanation for assertion because the integral of e(-x2) cannot be expressed in terms of elementary functions, making it impossible to find by inspection.
440 Easy
440 of 995
Application +4 | -1
∫(ex+e−x)2⋅(ex−e−x)dx\int\left(e^x+e^{-x}\right)^2 \cdot\left(e^x-e^{-x}\right) d x is equal to
View answer and explanation
Correct option: D
13(ex+e−x)3+C\frac{1}{3}\left(e^x+e^{-x}\right)^3+C
Option D (13(ex+e−x)3+C\frac{{1}}{{3}}\left(e^x+e^{-x}\right)^3+C) is correct because by substituting t=ex+e−xt = e^x + e^{-x}, the integral simplifies to 13t3+C\frac{{1}}{{3}}t^3 + C.
441 Easy
441 of 995
Application +4 | -1
∫  e3  log⁡x(x4+1)−1  dx\int \,\,{e^{3\,\,\log x}}{\left( {{x^4} + 1} \right)^{ - 1}}\,\,dx is equal to
View answer and explanation
Correct option: B
14  log⁡  (x4+1)+  c\frac{1}{4}\,\,\log \,\,\left( {{x^4} + 1} \right) + \,\,c
Option B (14,,log⁡,(x4+1)+,c\frac{1}{4}\\,,\log \\,\left( {{{{x^4} + 1}}} \right) + \\,c) is correct because it is the result of simplifying the integral and applying a u-substitution.
442 Easy
442 of 995
Application +4 | -1
The value of ∫cos⁡xsin⁡x+cos⁡x dx\int \frac{\cos x}{\sin x+\cos x} \mathrm{~d} x is
View answer and explanation
Correct option: B
12(x+log⁡∣sin⁡x+cos⁡x∣)+C\frac{1}{2}(x+\log |\sin x+\cos x|)+C
Option B is correct because the integral can be solved by splitting the numerator and using logarithmic integration.
443 Easy
443 of 995
Application +4 | -1
∫  dx2 x  (1+x)\int \,\,\frac{{dx}}{{2\,\sqrt x \,\,\left( {1 + x} \right)}} is equal to
View answer and explanation
Correct option: B
tan⁡−1  (x)+c{\tan ^{ - 1}}\,\,\left( {\sqrt x } \right) + c
Option B (tan⁡−1  (x)+c{\tan ^{ - 1}}\,\,\left( {\sqrt x } \right) + c) is correct because it is obtained by using a simple u-substitution with u=xu = \sqrt{x}.
444 Easy
444 of 995
Analyzing +4 | -1
If ∫1x+x5dx=f(x)+C\int \frac{1}{x+x^5} d x=f(x)+C, then the value of ∫x4x+x5dx\int \frac{x^4}{x+x^5} d x is
View answer and explanation
Correct option: A
log⁡x−f(x)+C\log x-f(x)+C
The correct answer is A (log⁡x−f(x)+C\log x-f(x)+C) because by manipulating the integrand and using the given information, the integral can be expressed in terms of f(x)f(x) and log⁡x\log x.
445 Medium
445 of 995
Application +4 | -1
If ∫  [log⁡ (log⁡  x)  +  1(log⁡  x)2]dx=x  [ f(x)  −  g(x) ]  +  c\int \,\,\left[ {\log \,(\log \,\,x)\,\, + \,\,\frac{1}{{{{\left( {\log \,\,x} \right)}^2}}}} \right]dx = x\,\,\left[ {\,f\left( x \right)\,\, - \,\,\mathrm{g}\left( x \right)\,} \right]\,\, + \,\,c, then
View answer and explanation
Correct option: A
f  (x)=log⁡ (log⁡  x);  g(x)=1log⁡  xf\,\,\left( x \right) = \log \,(\log \,\,x);\,\,\mathrm{g}\left( x \right) = \frac{1}{{\log \,\,x}}
Option A (f  (x)=log⁡ (log⁡  x);  g(x)=1log⁡  xf\,\,\left( x \right) = \log \,(\log \,\,x);\,\,\mathrm{g}\left( x \right) = \frac{1}{{\log \,\,x}}) is correct because after integrating by parts, the resulting expression matches the given form with these functions.
446 Easy
446 of 995
Application +4 | -1
∫{x+x2+1}nx2+1dx\int \frac{\left\{x+\sqrt{x^2+1}\right\}^n}{\sqrt{x^2+1}} d x is equal to
View answer and explanation
Correct option: B
1n{x+x2+1}n+C\frac{1}{n}\left\{x+\sqrt{x^2+1}\right\}^n+C
Option B is correct. The integral is solved by recognizing that the derivative of x+x2+1x + \sqrt{x^2 + 1} is x+x2+1x2+1\frac{{x + \sqrt{x^2 + 1}}}{{\sqrt{x^2 + 1}}}, allowing for a direct substitution and integration.
447 Easy
447 of 995
Application +4 | -1
∫1+x4(1−x4)3/2dx\int \frac{1+x^4}{\left(1-x^4\right)^{3 / 2}} d x is equal to
View answer and explanation
Correct option: B
11x2−x2+C\frac{1}{\sqrt{\frac{1}{x^2}-x^2}}+C
Option B (11x2−x2+C\frac{{1}}{{\sqrt{{\frac{{1}}{{x^2}}-x^2}}}}+C) is correct because it is the result of the integration after appropriate substitution and simplification.
448 Easy
448 of 995
Application +4 | -1
∫  dxcos⁡  x  +  3  sin⁡  x\int \,\,\frac{{dx}}{{\cos \,\,x\,\, + \,\,\sqrt 3 \,\,\sin \,\,x}} equals
View answer and explanation
Correct option: A
12  log⁡  tan⁡  (x2  +  π12)+c\frac{1}{2}\,\,\log \,\,\tan \,\,\left( {\frac{x}{2}\,\, + \,\,\frac{\pi }{{12}}} \right) + c
Option A is correct because the integral of the given function simplifies to a logarithmic tangent function with the correct phase shift.
449 Medium
449 of 995
Application +4 | -1
If ∫  xa3−x3  dx=g(x)+c\int \,\,\sqrt {\frac{x}{{{a^3} - {x^3}}}} \,\,dx = \mathrm{g}\left( x \right) + c, then g(x) is equal to
View answer and explanation
Correct option: C
23  sin⁡−1  (x3a3)\frac{2}{3}\,\,{\sin ^{ - 1}}\,\,\left( {\sqrt {\frac{{{x^3}}}{{{a^3}}}} } \right)
Option C is correct because the integral can be solved using trigonometric substitution, leading to an inverse sine function.
450 Easy
450 of 995
Application +4 | -1
∫  axx  dx\int \,\,\frac{{{a^{\sqrt x }}}}{{\sqrt x }}\,\,dx is equal to
View answer and explanation
Correct option: B
2axlog⁡  a  +  C\frac{{2{a^{\sqrt x }}}}{{\log \,\,a}}\,\, + \,\,C
The correct answer is B (2axlog⁡  a +  C\frac{{2{a^{\sqrt x }}}}{{\log \,\,a}}\, + \,\,C) because after applying the substitution method and integrating, the result matches option B.
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