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JEE • MATHEMATICS • INDEFINITE INTEGRATION

Indefinite integration practice · Mixed difficulty

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Showing 151-180 of 995 questions

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151 Easy
151 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 equals to
View answer and explanation
Correct option: D
None of these
The correct answer is D (None of these) because the result of the integration is 12(x2−1)ex2+C\frac{{1}}{{2}}(x^2-1) e^{x^2}+C, which does not match any of the provided options.
152 Medium
152 of 995
Analyzing +4 | -1
∫  (x+1)3/2  [sin⁡−1  (log⁡  x)  +  cos⁡−1  (log⁡  x)]  dx\int \,\,{\left( {x + 1} \right)^{3/2}}\,\,[{\sin ^{ - 1}}\,\,(\log \,\,x)\,\, + \,\,{\cos ^{ - 1}}\,\,(\log \,\,x)]\,\,dx is equal to
View answer and explanation
Correct option: C
Does not exist
Option C (Does not exist) is correct because the domain of the inverse trigonometric functions requires the argument to be within [-1, 1], which is not satisfied for all x in the integral.
153 Medium
153 of 995
Analyzing +4 | -1
∫Sin⁡−1x−Cos⁡−1xSin⁡−1x+Cos⁡−1xdx=\int \frac{\operatorname{Sin}^{-1} \sqrt{x}-\operatorname{Cos}^{-1} \sqrt{x}}{\operatorname{Sin}^{-1} \sqrt{x}+\operatorname{Cos}^{-1} \sqrt{x}} d x=
View answer and explanation
Correct option: A
4π[x−x22+xSin⁡−1x−12Sin⁡−11−x]−x+c\frac{4}{\pi}\left[\frac{\sqrt{x-x^2}}{2}+x \operatorname{Sin}^{-1} \sqrt{x}-\frac{1}{2} \operatorname{Sin}^{-1} \sqrt{1-x}\right]-x+c
The correct answer is A (4π[x−x22+xSin⁡−1x−12Sin⁡−11−x]−x+c\frac{4}{\pi}\left[\frac{\sqrt{x-x^2}}{2}+x \operatorname{Sin}^{-1} \sqrt{x}-\frac{1}{2} \operatorname{Sin}^{-1} \sqrt{1-x}\right]-x+c) because after simplifying the integral using trigonometric identities and integration by parts, the result matches option A.
154 Medium
154 of 995
Analyzing +4 | -1
Let x2≠n  π−1,   n∈N{x^2} \ne n\;\pi - 1,\;\,n \in N. The indefinite integral of
x  2  sin⁡  (x2+  1)  −  sin⁡  2(x2+  1)2  sin⁡  (x2+  1)  +  sin⁡  2(x2+  1)x\,\,\sqrt {\frac{{2\,\,\sin \,\,\left( {{x^2} + \,\,1} \right)\,\, - \,\,\sin \,\,2\left( {{x^2} + \,\,1} \right)}}{{2\,\,\sin \,\,\left( {{x^2} + \,\,1} \right)\,\, + \,\,\sin \,\,2\left( {{x^2} + \,\,1} \right)}}}

is but for an arbitrary constant
View answer and explanation
Correct option: B
log⁡  ∣ 12  sec⁡  (x2+12) ∣\log \,\,\left| {\,\frac{1}{2}\,\,\sec \,\,\left( {\frac{{{x^2} + 1}}{2}} \right)\,} \right|
Option B (\log \,\left| {\\,\frac{1}{2}\\,\sec \\,\left( {\frac{{{{{x^2} + 1}}}}{2}} \right)\} \right|\log \,\left| {\\,\frac{1}{2}\\,\sec \\,\left( {\frac{{{{{x^2} + 1}}}}{2}} \right)\} \right|) is correct because it is the result of the indefinite integral after simplification and substitution.
155 Medium
155 of 995
Analyzing +4 | -1
∫cos⁡2θlog⁡e(cos⁡θ+sin⁡θcos⁡θ−sin⁡θ)dθ=sin⁡2θ2×log⁡e(cos⁡θ+sin⁡θcos⁡θ−sin⁡θ)−12f(x)+c\int \cos 2 \theta \log _e\left(\frac{\cos \theta+\sin \theta}{\cos \theta-\sin \theta}\right) d \theta=\frac{\sin 2 \theta}{2}\times \log _e\left(\frac{\cos \theta+\sin \theta}{\cos \theta-\sin \theta}\right)-\frac{1}{2} f(x)+c where f(x)f(x) is
View answer and explanation
Correct option: B
log⁡e(sec⁡2θ)\log _e(\sec 2 \theta)
The correct answer is B (log⁡e(sec⁡2θ)\log _e(\sec 2 \theta)) because using integration by parts, the integral simplifies, and the remaining integral is a standard form that results in the logarithm of secant.
156 Easy
156 of 995
Application +4 | -1
∫32  x3(log⁡  x)2dx\int 32\,\,{x^3}{\left( {\log \,\,x} \right)^2}dx is equal to
View answer and explanation
Correct option: B
x4  { 8(log⁡  x)2− 4  log⁡  x+1 }+c{x^4}\,\,\left\{ {\,8{{\left( {\log \,\,x} \right)}^2} - \,4\,\,\log \,\,x + 1\,} \right\} + c
Option B (x4  { 8(log⁡ x)2− 4  log⁡ x+1 }+c{x^4}\,\,\{\,8{{\left( {\log \,x} \right)}^2} - \,4\,\,\log \,x + 1\,\} + c) is correct because it is the result of applying integration by parts twice to the given integral.
157 Hard
157 of 995
Analyzing +4 | -1
If
∫x−11(1+x4)−1/2dx=−110x−a(1+x4)5/2+13x−b(1+x4)3/2+−12x−c(1+x4)1/2+k\int x^{-11}\left(1+x^4\right)^{-1 / 2} d x= -\frac{1}{10} x^{-a}\left(1+x^4\right)^{5 / 2}+\frac{1}{3} x^{-b}\left(1+x^4\right)^{3 / 2} +\frac{-1}{2} x^{-c}\left(1+x^4\right)^{1 / 2}+k
then a+b+ca+b+c equals
View answer and explanation
Correct option: D
18
Option D (18) is correct because the values of a, b, and c are determined by solving the integral and comparing coefficients, and their sum is 18.
158 Medium
158 of 995
Application +4 | -1
∫sec⁡xdxcos⁡2x=\int \frac{\sec x d x}{\sqrt{\cos 2 x}}=
View answer and explanation
Correct option: A
sin⁡−1(tan⁡x)\sin ^{-1}(\tan x)
Option A is correct because the integral can be rewritten and solved using a trigonometric substitution, resulting in sin⁡−1(tan⁡x)\sin^{-1}(\tan x).
159 Easy
159 of 995
Application +4 | -1
∫ex(1+tan⁡x)sec⁡xdx=\int e^x(1+\tan x) \sec x d x=
View answer and explanation
Correct option: C
exsec⁡xe^x \sec x
Option C is correct because the integral can be simplified to the form ∫ex[f(x)+f′(x)]dx\int e^x[f(x) + f'(x)] dx, where f(x)=sec⁡xf(x) = \sec x and f′(x)=sec⁡xtan⁡xf'(x) = \sec x \tan x.
160 Easy
160 of 995
Application +4 | -1
∫x3ex2dx=\int x^3 e^{x^2} d x=
View answer and explanation
Correct option: C
12(x2−1)ex2+c\frac{1}{2}\left(x^2-1\right) e^{x^2}+c
Option C is correct because the integral of x3 e(x2) dx is found using substitution and integration by parts, resulting in 1/2(x2-1)e(x2) + c.
161 Medium
161 of 995
Analyzing +4 | -1
∫a+bsin⁡x(b+asin⁡x)2dx\int\frac{a+b \sin x}{(b+a \sin x)^2} d x equals
View answer and explanation
Correct option: A
−(acos⁡xb+asin⁡x)+c-\left(\frac{a \cos x}{b+a \sin x}\right)+c
Option A (−(acos⁡xb+asin⁡x)+c-\left(\frac{{a \cos x}}{{b+a \sin x}}\right)+c) is correct because the integral simplifies to this expression after algebraic manipulation and recognizing the derivative of a function.
162 Medium
162 of 995
Application +4 | -1
∫(x+2x+4)2exdx\int\left(\frac{x+2}{x+4}\right)^2 e^x d x is equal to
View answer and explanation
Correct option: A
ex(xx+4)+ce^x\left(\frac{x}{x+4}\right)+c
Option A (ex(xx+4)+ce^x\left(\frac{{x}}{{x+4}}\right)+c) is correct because the integral can be solved by simplifying the expression and applying the rule ∫ex[f(x)+f′(x)]dx=exf(x)+c\int e^x [f(x) + f'(x)] dx = e^x f(x) + c.
163 Easy
163 of 995
Application +4 | -1
∫dx(x2+1)(x2+4)=\int \frac{d x}{\left(x^2+1\right)\left(x^2+4\right)}=
View answer and explanation
Correct option: C
13tan⁡−1x−16tan⁡−1x2+c\frac{1}{3} \tan ^{-1} x-\frac{1}{6} \tan ^{-1} \frac{x}{2}+c
Option C, 13tan⁡−1x−16tan⁡−1x2+c\frac{1}{3} \tan ^{-1} x-\frac{1}{6} \tan ^{-1} \frac{x}{2}+c, is the correct answer because it results from partial fraction decomposition and integration of the given expression.
164 Easy
164 of 995
Application +4 | -1
∫32  x3(log⁡  x)2    dx\int 32\,\,{x^3}{(\log \,\,x)^2}\,\,\;dx is equal to
View answer and explanation
Correct option: B
x4 { 8 (log⁡  x)2−  4 log⁡  x+1 }+ c{x^4}\,\{ \,8\,{(\log \,\,x)^2} - \,\,4\,\log \,\,x + 1\,\} + \,c
Option B is correct because the integral is solved using integration by parts twice.
165 Medium
165 of 995
Application +4 | -1
∫dxcos⁡x+3sin⁡x\int \frac{d x}{\cos x+\sqrt{3} \sin x} equals
View answer and explanation
Correct option: C
12log⁡tan⁡(x2+π12)+C\frac{1}{2} \log \tan \left(\frac{\mathrm{x}}{2}+\frac{\pi}{12}\right)+\mathrm{C}
Option C (12log⁡tan⁡(x2+π12)+C\frac{1}{2} \log \tan \left(\frac{\mathrm{x}}{2}+\frac{\pi}{12}\right)+\mathrm{C}) is correct because the integral can be simplified using trigonometric identities and the integral of cosecant.

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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
166 Easy
166 of 995
Application +4 | -1
∫  cosec4  x    dx\int \,\,\mathrm{cose}{\mathrm{c}^4}\,\,x\,\,\;dx is equal to
View answer and explanation
Correct option: C
−  cot⁡  x  −  cot⁡3  x3  +  c- \,\,\cot \,\,x\,\, - \,\,\frac{{{{\cot }^3}\,\,x}}{3}\,\, + \,\,c
Option C (− cot⁡ x − cot⁡3 x3 + c- \,\cot \,x\, - \,\frac{{{{\cot }^3}\,x}}{3}\, + \,c) is correct because it is the result of integrating cosec4(x)cosec^4(x) by rewriting it as cosec2(x)(1+cot2(x))cosec^2(x)(1 + cot^2(x)).
167 Easy
167 of 995
Application +4 | -1
∫dxx(x  +  1)\int \frac{{dx}}{{x\left( {x\,\, + \,\,1} \right)}} equals
Where cc is an arbitrary constant
View answer and explanation
Correct option: B
log⁡  ∣ xx  +  1 ∣+c\log \,\,\left| {\,\frac{x}{{x\,\, + \,\,1}}\,} \right| + c
Option B, \log \,\,\left| {\,\frac{x}{{x\,\,+\,\,1}}\} \right| + c\log \,\,\left| {\,\frac{x}{{x\,\,+\,\,1}}\} \right| + c, is correct because the integral of the given expression can be solved using partial fractions, resulting in this logarithmic form.
168 Medium
168 of 995
Application +4 | -1
∫{log⁡x−11+(log⁡x)2}2dx\int\left\{\frac{\log x-1}{1+(\log x)^2}\right\}^2 d x 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 after substitution and integration by parts, the integral simplifies to this expression.
169 Medium
169 of 995
Analyzing +4 | -1
∫(cos⁡x−cos⁡3x1−cos⁡3x)1/2dx=\int\left(\frac{\cos x-\cos ^3 x}{1-\cos ^3 x}\right)^{1 / 2} d x=
View answer and explanation
Correct option: B
−23Sin⁡−1(cos⁡3/2x)+c-\frac{2}{3} \operatorname{Sin}^{-1}\left(\cos ^{3 / 2} x\right)+c
Option C (13sin⁡3x−sin⁡x\frac{1}{3} \sin ^3 x-\sin x) is correct because after integrating and comparing with the given form, g(x) matches this expression.
170 Medium
170 of 995
Application +4 | -1
∫x3x2+2dx=\int \frac{x^3}{\sqrt{x^2+2}} d x=
View answer and explanation
Correct option: B
13(x2+2)3/2−2(x2+2)1/2+c\frac{1}{3}\left(x^2+2\right)^{3 / 2}-2\left(x^2+2\right)^{1 / 2}+c
Option B is correct because the integral is solved using a substitution, resulting in 13(x2+2)3/2−2(x2+2)1/2+c\frac{1}{3}(x^2+2)^{3/2} - 2(x^2+2)^{1/2} + c.
171 Easy
171 of 995
Understanding +4 | -1
If d  [ f  (x)]=etan⁡x  sec⁡2  x    dx,    then    f(x)d\,\,\left[ {\,f\,\,\left( x \right)} \right] = {e^{\tan x}}\,\,{\sec ^2}\,\,x\,\,\;dx,\;\,\,\mathrm{then}\;\,\,f\left( x \right) is equal to

View answer and explanation
Correct option: A
etan⁡x  +  c{e^{\tan x}}\,\, + \,\,c
Option A (e(tan x) + c) is correct because it is the antiderivative of e(tan x) * sec2(x), found by recognizing that sec2(x) is the derivative of tan(x).
172 Medium
172 of 995
Application +4 | -1
The value of the integral ∫03αcosec(x−α)cosec(x−2α)dx,\int\limits_0^{3\alpha } {{\text{cosec}}\left( {x - \alpha } \right){\text{cosec}}\left( {x - 2\alpha } \right)dx} , is
View answer and explanation
Correct option: D
2    cosec    α  log⁡  (12  sec⁡  α)2\;\,\,\mathrm{cosec}\,\,\;\alpha \,\,\log \,\,\left( {\frac{1}{2}\,\,\sec \,\,\alpha } \right)
The correct answer is D (2    cosec   α  log⁡  (12  sec⁡  α)2\;\,\,\mathrm{cosec}\,\;\alpha \,\,\log \,\,\left( {\frac{{1}}{{2}}\,\,\sec \,\,\alpha } \right)) because it is the result of evaluating the definite integral using trigonometric identities and logarithmic properties.
173 Easy
173 of 995
Application +4 | -1
∫x2ex3cos⁡(ex3)dx\int x^2 e^{x^3} \cos \left(e^{x^3}\right) d x is equal to
View answer and explanation
Correct option: C
13sin⁡(ex3)+C\frac{1}{3} \sin \left(e^{x^3}\right)+C
Option C (13sin⁡(ex3)+C\frac{1}{3} \sin \left(e^{x^3}\right)+C) is correct because after simplifying the integral using substitution, the result matches this option.
174 Medium
174 of 995
Application +4 | -1
The value of the integral ∫0∞11  +  x4dx\int \limits_0^\infty \frac{1}{{1\,\, + \,\,{x^4}}}dx, is
View answer and explanation
Correct option: C
π2 2\frac{\pi }{{2\,\sqrt 2 }}
Option C (π22\frac{\pi}{{2\sqrt 2 }}) is correct because the integral evaluates to this value after using trigonometric substitution and simplification.
175 Medium
175 of 995
Application +4 | -1
If ∫log⁡(x+1+x2)1+x2 dx=gof(x)+\int \frac{\log \left(x+\sqrt{1+x^2}\right)}{\sqrt{1+x^2}} \mathrm{~d} x=g o f(x)+ Const. then
View answer and explanation
Correct option: C
f(x)=log⁡(x+x2+1)f(x)=\log \left(x+\sqrt{x^2+1}\right) and g(x)=x2/2g(x)=x^2 / 2
The correct answer is C (f(x)=log⁡(x+x2+1)f(x)=\log (x+\sqrt{x^2+1}) and g(x)=x2/2g(x)=x^2 / 2) because after substituting t=log⁡(x+1+x2)t = \log(x + \sqrt{1+x^2}), the integral becomes ∫tdt=t2/2+C\int t dt = t^2/2 + C, which matches the given form.
176 Easy
176 of 995
Application +4 | -1
The value of ∫  x21  +  x6  dx\int \,\,\frac{{{x^2}}}{{1\,\, + \,\,{x^6}}}\,\,dx is
View answer and explanation
Correct option: B
13  tan⁡−1  (x3)  +  c\frac{1}{3}\,\,{\tan ^{ - 1}}\,\,\left( {{x^3}} \right)\,\, + \,\,c
Option B (13 tan⁡−1 (x3) + c\frac{1}{3}\,{\tan ^{ - 1}}\,\left( {{x^3}} \right)\, + \,c) is correct because it results from a u-substitution where u=x3u = x^3, leading to the integral of 11+u2\frac{1}{1+u^2}.
177 Easy
177 of 995
Application +4 | -1
∫e−x1+exdx=\int \frac{e^{-x}}{1+e^x} d x=
View answer and explanation
Correct option: A
log⁡(1+ex)−x−e−x+c\log \left(1+e^x\right)-x-e^{-x}+c
Option A is correct because the integral can be solved by rewriting the integrand and using substitution, leading to the expression log⁡(1+ex)−x−e−x+c\log(1+e^x) - x - e^{-x} + c.
178 Medium
178 of 995
Application +4 | -1
Let f(x)=∫2x(x2+1)(x2+3)dxf(x)=\int \frac{2 x}{\left(x^{2}+1\right)\left(x^{2}+3\right)} d x. If f(3)=12(log⁡e5−log⁡e6)f(3)=\frac{1}{2}\left(\log _{e} 5-\log _{e} 6\right), then f(4)f(4) is equal to
View answer and explanation
Correct option: C
12(log⁡e17−log⁡e19)\frac{1}{2}\left(\log _{e} 17-\log _{e} 19\right)
Option C (12(log⁡e17−log⁡e19)\frac{{1}}{{2}}(\log _{{e}} 17-\log _{{e}} 19)) is correct because after evaluating the integral and using the given condition f(3), we can find the value of f(4).
179 Medium
179 of 995
Analyzing +4 | -1
If ∫dx(x2−2x+10)2=A(tan⁡−1(x−13)+f(x)x2−2x+10)+C\int \frac{d x}{\left(x^{2}-2 x+10\right)^{2}}=A\left(\tan ^{-1}\left(\frac{x-1}{3}\right)+\frac{f(x)}{x^{2}-2 x+10}\right)+C where C\mathrm{C} is a constant of integration then :
View answer and explanation
Correct option: B
A=154A=\frac{1}{54} and f(x)=3(x−1)f(x)=3(x-1)
Option B (A=154A=\frac{{1}}{{54}} and f(x)=3(x−1)f(x)=3(x-1)) is correct because after solving the integral using trigonometric substitution, the result matches the given form with the specified values for A and f(x).
180 Medium
180 of 995
Application +4 | -1
∫cos⁡(log⁡ex)dx\int \cos \left(\log _e x\right) d x is equal to
View answer and explanation
Correct option: A
12x{cos⁡(log⁡ex)+sin⁡(log⁡ex)}\frac{1}{2} x\left\{\cos \left(\log _e x\right)+\sin \left(\log _e x\right)\right\}
Option A is correct because the integral of cos⁡(log⁡ex)\cos(\log_e x) is found using integration by parts twice, resulting in the given expression.
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