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JEE • MATHEMATICS • INDEFINITE INTEGRATION

Indefinite integration practice · Mixed difficulty

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961 Easy
961 of 995
Application +4 | -1
The integral ∫dxx+x23\int \frac{d x}{\sqrt{x}+\sqrt[3]{x^2}} represents the function
View answer and explanation
Correct option: B
3x3−6x6+6log⁡∣1+x6∣+c3 \sqrt[3]{x}-6 \sqrt[6]{x}+6 \log |1+\sqrt[6]{x}|+c
Option B (3x3−6x6+6log⁡∣1+x6∣+c3 \sqrt[3]{x}-6 \sqrt[6]{x}+6 \log |1+\sqrt[6]{x}|+c) is correct because it is the result of integrating the given expression using substitution.
962 Easy
962 of 995
Application +4 | -1
If ddxf(x)=xcos⁡x+sin⁡x\frac{d}{d x} f(x)=x \cos x+\sin x and f(0)=2f(0)=2, then f(x)=f(x)=
View answer and explanation
Correct option: C
xsin⁡x+2x \sin x+2
Option C is correct because integrating x cos(x) + sin(x) results in x sin(x) + C, and using the initial condition f(0) = 2, we find that C = 2, so f(x) = x sin(x) + 2.
963 Easy
963 of 995
Application +4 | -1
∫cos⁡3  x . elog⁡sin⁡x  dx\int {\cos ^3}\,\,x\,.\,{e^{\log \sin x}}\,\,dx is equal to
View answer and explanation
Correct option: B
−  cos⁡4  x4  +  c- \,\,\frac{{{{\cos }^4}\,\,x}}{4}\,\, + \,\,c
Option B, −  cos⁡4  x4  +  c- \,\,\frac{{{{\cos }^4}\,\,x}}{4}\,\, + \,\,c, is correct because after simplifying the expression and using substitution, the integral evaluates to this result.
964 Easy
964 of 995
Application +4 | -1
The value of the integral ∫xsin⁡x2esec⁡x2cos⁡2x2dx\int \frac{x \sin x^2 e^{\sec x^2}}{\cos ^2 x^2} d x, is
View answer and explanation
Correct option: A
12esec⁡x2+C\frac{1}{2} e^{\sec x^2}+C
Option A is correct because the integral is solved by recognizing the derivative of the secant term and applying a simple substitution.
965 Easy
965 of 995
Application +4 | -1
∫  ax/2a−x−  ax  dx=\int \,\,\frac{{{a^{x/2}}}}{{\sqrt {{a^{ - x}} - \,\,{a^x}} }}\,\,dx =
View answer and explanation
Correct option: A
1log⁡  a  sin⁡−1  (ax)\frac{1}{{\log \,\,a}}\,\,{\sin ^{ - 1}}\,\,\left( {{a^x}} \right)
Option A (1log⁡ a sin⁡−1 (ax)\frac{1}{{\log \,a}}\,{\sin ^{ - 1}}\,\left( {{a^x}} \right)) is correct because it is the result of integrating the given expression using substitution.
966 Easy
966 of 995
Application +4 | -1
∫1cos⁡x(1+cos⁡x)dx=\int \frac{1}{\cos x(1+\cos x)} d x=
View answer and explanation
Correct option: D
log⁡(sec⁡x+tan⁡x)−tan⁡x2+c\log (\sec x+\tan x)-\tan \frac{x}{2}+c
Option D (log⁡(sec⁡x+tan⁡x)−tan⁡x2+c\log (\sec x+\tan x)-\tan \frac{x}{2}+c) is correct because the integral is solved using trigonometric identities and simplification.
967 Medium
967 of 995
Analyzing +4 | -1
Integration of 1x2+  9\frac{1}{{\sqrt {{x^2} + \,\,9} }} with respect to (x2+  1)\left( {{x^2} + \,\,1} \right) is equal to
View answer and explanation
Correct option: C
2 x2+  9  +  C2\,\sqrt {{x^2} + \,\,9} \,\, + \,\,C
The correct answer is C (2,x2+  9  +  C2\\,\sqrt {{{{x^2} + \,\,9}}} \,\, + \,\,C) because the integral is solved by recognizing the differential and integrating.
968 Medium
968 of 995
Analyzing +4 | -1
∫ex2(1−x2)(1−x2)1−2x2dx\int \frac{e^{x \sqrt{2}}\left(1-x^2\right)}{(1-x \sqrt{2}) \sqrt{1-2 x^2}} d x is equal to
View answer and explanation
Correct option: C
122ex21+x21−x2+c\frac{1}{2 \sqrt{2}} e^{x \sqrt{2}} \sqrt{\frac{1+x \sqrt{2}}{1-x \sqrt{2}}}+c
The correct answer is C (122ex21+x21−x2+c\frac{1}{2 \sqrt{2}} e^{x \sqrt{2}} \sqrt{\frac{1+x \sqrt{2}}{1-x \sqrt{2}}}+c) because after substitution and simplification, the integral can be solved, and the result matches option C.
969 Easy
969 of 995
Application +4 | -1
∫ex(x2+1)(x+1)2dx=\int e^x \frac{\left(x^2+1\right)}{(x+1)^2} d x=
View answer and explanation
Correct option: A
(x−1x+1)ex+c\left(\frac{x-1}{x+1}\right) e^x+c
Option A is correct because the integral can be expressed in the form ∫ex[f(x)+f′(x)]dx\int e^x [f(x) + f'(x)] dx, where f(x)=x−1x+1f(x) = \frac{x-1}{x+1} and f′(x)=2(x+1)2f'(x) = \frac{2}{(x+1)^2}.
970 Easy
970 of 995
Understanding +4 | -1
If ∫f(x)dx=f(x)\int f(x) d x=f(x), then
View answer and explanation
Correct option: D
f(x)=exf(x)=e^x
Option D (f(x)=exf(x)=e^x) is correct because the integral of exe^x is exe^x (plus a constant, which we can ignore in this context).
971 Easy
971 of 995
Application +4 | -1
∫dxsin⁡  x−cos⁡  x+2\int \frac{{dx}}{{\sin \,\,x - \cos \,\,x + \sqrt 2 }} equals
View answer and explanation
Correct option: D
−12  cot⁡  (x2  +  π8)+c- \frac{1}{{\sqrt 2 }}\,\,\cot \,\,\left( {\frac{x}{2}\,\, + \,\,\frac{\pi }{8}} \right) + c
The correct answer is D (−12  cot⁡  (x2  +  π8)+c- \frac{1}{{\sqrt 2 }}\,\,\cot \,\,\left( {\frac{x}{2}\,\, + \,\,\frac{\pi }{8}} \right) + c) because after applying trigonometric identities and integrating, the result matches option D.
972 Easy
972 of 995
Understanding +4 | -1
∫sin⁡2x1+sin⁡2xdx=\int \frac{\sin 2 x}{1+\sin ^2 x} d x=
View answer and explanation
Correct option: B
log⁡(1+sin⁡2x)+c\log \left(1+\sin ^2 x\right)+c
Option B (log⁡(1+sin⁡2x)+c\log \left(1+\sin ^2 x\right)+c) is correct because the derivative of 1+sin⁡2x1+\sin^2 x is sin⁡2x\sin 2x, which is present in the integrand.
973 Easy
973 of 995
Analyzing +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 is correct because after dividing by x2x^2 and substituting x+1x=tx + \frac{1}{x} = t, the integral simplifies to 12sec⁡−1(x2+12x)+C\frac{1}{\sqrt{2}} \sec ^{-1}\left(\frac{x^2+1}{\sqrt{2} x}\right)+C.
974 Medium
974 of 995
Analyzing +4 | -1
Given, f(x)=∣0x2−sin⁡xcos⁡x−2sin⁡x−x201−2x2−cos⁡x2x−10∣f(x)=\left|\begin{array}{ccc}0 & x^2-\sin x & \cos x-2 \\ \sin x-x^2 & 0 & 1-2 x \\ 2-\cos x & 2 x-1 & 0\end{array}\right|, then ∫f(x)dx\int f(x) d x is equal to
View answer and explanation
Correct option: D
None of the above
The correct answer is D (None of the above) because the determinant is zero, making the integral zero as well.
975 Easy
975 of 995
Analyzing +4 | -1
∫e(x2+4ln⁡x)−x3ex2x−1dx\int \frac{e^{\left(x^2+4 \ln x\right)}-x^3 e^{x^2}}{x-1} d x is equal to
View answer and explanation
Correct option: D
None of these
Option D (None of these) is correct because the integral simplifies to 12(x2−1)ex2+C\frac{1}{2}(x^2-1) e^{x^2}+C, which is not any of the given options.

Explore Other Chapters in MATHEMATICS

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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
976 Easy
976 of 995
Application +4 | -1
∫dx(x−a)3/2(x+a)1/2\int \frac{d x}{(x-a)^{3 / 2}(x+a)^{1 / 2}} is equal to
View answer and explanation
Correct option: D
−1ax+ax−a+c-\frac{1}{a} \sqrt{\frac{x+a}{x-a}}+c
Option D (−1ax+ax−a+c-\frac{1}{a} \sqrt{\frac{x+a}{x-a}}+c) is correct because after substitution and integration, the expression simplifies to this form.
977 Medium
977 of 995
Application +4 | -1
Evaluate: ∫sin⁡(x−a)sin⁡(x+a)dx\int \frac{\sin (x-a)}{\sin (x+a)} d x
View answer and explanation
Correct option: C
(x+a)cos⁡2a−sin⁡2alog⁡∣sin⁡(x+a)∣+c(x+a) \cos 2 a-\sin 2 a \log |\sin (x+a)|+c
Option C ((x+a)cos⁡2a−sin⁡2alog⁡∣sin⁡(x+a)∣+c(x+a) \cos 2 a-\sin 2 a \log |\sin (x+a)|+c) is correct because the integral is solved by substituting x+a=tx+a=t and using trigonometric identities.
978 Medium
978 of 995
Application +4 | -1
∫  dx(a2+x2)3/2\int \,\,\frac{{dx}}{{{{\left( {{a^2} + {x^2}} \right)}^{3/2}}}} is equal to
View answer and explanation
Correct option: B
xa2(a2+x2)1/2+ c\frac{x}{{{a^2}{{\left( {{a^2} + {x^2}} \right)}^{1/2}}}} + \,c
Option B is correct because after trigonometric substitution and simplification, the integral evaluates to xa2(a2+x2)1/2+ c\frac{x}{{{{a^2}{{\left( {{a^2} + {x^2}} \right)}^{1/2}}}}} + \,c.
979 Medium
979 of 995
Application +4 | -1
∫  x1  +  x34  dx\int \,\,\frac{{\sqrt x }}{{1\,\, + \,\,\sqrt[4]{{{x^3}}}}}\,\,dx is equal to
View answer and explanation
Correct option: B
43  [1+x3/4  −  log⁡e  (1  +  x3/4)]+C\frac{4}{3}\,\,\left[ {1 + {x^{3/4}}\,\, - \,\,{{\log }_e}\,\,\left( {1\,\, + \,\,{x^{3/4}}} \right)} \right] + C
Option B (43  [1+x3/4  −  log⁡e  (1  +  x3/4)]+C\frac{4}{3}\,\,\left[ {1 + {x^{3/4}}\,\, - \,\,{{\log }_e}\,\,\left( {1\,\, + \,\,{x^{3/4}}} \right)} \right] + C) is correct because after substitution and integration, the result matches this option.
980 Medium
980 of 995
Understanding +4 | -1
If nn is an odd positive integer, then ∫∣xn∣dx\int\left|x^n\right| d x is equal to
View answer and explanation
Correct option: C
∣x∣nxn+1+C\frac{|x|^n x}{n+1}+C
Option C (∣x∣nxn+1+C\frac{|x|^n x}{n+1}+C) is correct because it accounts for both positive and negative values of xx when nn is an odd positive integer.
981 Easy
981 of 995
Analyzing +4 | -1
∫x2(xsin⁡x+cos⁡x)2dx\int \frac{x^2}{(x \sin x+\cos x)^2} d x would be equal to
View answer and explanation
Correct option: B
sin⁡x−xcos⁡xxsin⁡x+cos⁡x+C\frac{\sin x-x \cos x}{x \sin x+\cos x}+C
Option B is correct because after applying integration by parts and simplifying, we get sin⁡x−xcos⁡xxsin⁡x+cos⁡x+C\frac{\sin x-x \cos x}{x \sin x+\cos x}+C.
982 Easy
982 of 995
Application +4 | -1
The value of the integral ∫  1+x21+x4  dx\int \,\,\frac{{1 + {x^2}}}{{1 + {x^4}}}\,\,dx is equal to
View answer and explanation
Correct option: B
12  tan⁡−1  (x2−  12x)\frac{1}{{\sqrt 2 }}\,\,{\tan ^{ - 1}}\,\,\left( {\frac{{{x^2} - \,\,1}}{{\sqrt 2 x}}} \right)
Option B (12,tan⁡−1(x2−,12x)\frac{1}{{\sqrt 2 }}\\,{\tan ^{ - 1}}\left( {\frac{{{{{x^2}} - \\,1}}}{{\sqrt 2 x}}} \right)) is correct because the integral is solved by dividing numerator and denominator by x2x^2 and then using a suitable substitution.
983 Easy
983 of 995
Application +4 | -1
∫tan⁡(log⁡x)xdx=\int \frac{\tan (\log x)}{x} d x=
View answer and explanation
Correct option: C
log⁡sec⁡(log⁡x)+c\log \sec (\log x)+c
Option C is correct because substituting t=log⁡xt = \log x transforms the integral to ∫tan⁡tdt\int \tan t dt, which equals log⁡∣sec⁡t∣+c=log⁡∣sec⁡(log⁡x)∣+c\log |\sec t| + c = \log |\sec(\log x)| + c.
984 Easy
984 of 995
Application +4 | -1
If ∫x7(1+x4)2 dx=14[log⁡(1+x4)+f(x)]+C\int \frac{x^7}{\left(1+x^4\right)^2} \mathrm{~d} x=\frac{1}{4}\left[\log \left(1+x^4\right)+f(x)\right]+C then
View answer and explanation
Correct option: C
f(x)=11+x4f(x)=\frac{1}{1+x^4}
The correct answer is f(x)=11+x4f(x)=\frac{1}{1+x^4} because after substituting t=1+x4t=1+x^4 and integrating, we can identify f(x)f(x) by comparing with the given expression.
985 Easy
985 of 995
Application +4 | -1
∫2(ex+e−x)2dx\int \frac{2}{\left(e^x+e^{-x}\right)^2} d x is equal to
View answer and explanation
Correct option: A
−e−xex+e−x+C\frac{-e^{-x}}{e^x+e^{-x}}+C
Option A (−e−xex+e−x+C\frac{-e^{-x}}{e^x+e^{-x}}+C) is correct because after simplifying the integral, the result matches this option.
986 Medium
986 of 995
Application +4 | -1
If ∫dxx−x3=Aln⁡∣x21−x2∣+c\int \frac{d x}{x-x^3}=A \ln \left|\frac{x^2}{1-x^2}\right|+c, then AA is equal to
View answer and explanation
Correct option: B
1/21/2
Option B (1/2) is correct because the integral simplifies to the given form with A = 1/2 after partial fraction decomposition.
987 Medium
987 of 995
Application +4 | -1
∫dxxlog⁡xlog⁡(log⁡x)=\int \frac{d x}{x \log x \log (\log x)}=
View answer and explanation
Correct option: B
log⁡[log⁡(log⁡x)]+c\log [\log (\log x)]+c
Option B (log⁡[log⁡(log⁡x)]+c\log [\log (\log x)]+c) is correct because it is the result of two u-substitutions.
988 Easy
988 of 995
Analyzing +4 | -1
The value of ∫(1+x)x(1+xex)2dx\int \frac{(1+x)}{x\left(1+x e^x\right)^2} d x, is equal to
View answer and explanation
Correct option: B
log⁡∣xex1+xex∣+11+xex+C\log \left|\frac{x e^x}{1+x e^x}\right|+\frac{1}{1+x e^x}+C
Option B is correct because after substituting 1+xex=t1+xe^x = t and using partial fractions, the integral evaluates to log⁡∣xex1+xex∣+11+xex+C\log \left|\frac{x e^x}{1+x e^x}\right|+\frac{1}{1+x e^x}+C.
989 Easy
989 of 995
Application +4 | -1
∫xx((ln⁡x)2+ln⁡x+1x)dx\int x^x\left((\ln x)^2+\ln x+\frac{1}{x}\right) d x is equal to:
View answer and explanation
Correct option: D
xxln⁡x+Cx^x \ln x+C
Option D (xxln⁡x+Cx^x \ln x+C) is correct because it results from correctly identifying the substitution and integrating the expression.
990 Medium
990 of 995
Application +4 | -1
∫dxcos⁡  x  −  sin⁡  x\int \frac{{dx}}{{\cos \,\,x\,\, - \,\,\sin \,\,x}} is equal to
View answer and explanation
Correct option: D
12  log⁡  ∣ tan⁡  (x2  +  3π8)+c ∣\frac{1}{{\sqrt 2 }}\,\,\log \,\,\left| {\,\tan \,\,\left( {\frac{x}{2}\,\, + \,\,\frac{{3\pi }}{8}} \right) + c\,} \right|
Option D is correct. The integral ∫dxcos⁡  x  −  sin⁡  x\int \frac{{dx}}{{\cos \,\,x\,\, - \,\,\sin \,\,x}} can be solved by rewriting the denominator and using trigonometric identities.
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