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JEE • MATHEMATICS • INDEFINITE INTEGRATION

Indefinite integration practice · Mixed difficulty

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Showing 661-690 of 995 questions

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661 Easy
661 of 995
Application +4 | -1
∫x31−x8dx=\int \frac{x^3}{\sqrt{1-x^8}} d x=
View answer and explanation
Correct option: C
14sin⁡−1(x4)+c\frac{1}{4} \sin ^{-1}\left(x^4\right)+c
Option C (14sin⁡−1(x4)+c\frac{1}{4} \sin ^{-1}\left(x^4\right)+c) is correct because it is the result of integrating the given expression using substitution.
662 Easy
662 of 995
Application +4 | -1
∫xtan⁡−1x(1+x2)3/2dx=\int \frac{x \tan ^{-1} x}{\left(1+x^2\right)^{3 / 2}} d x=
View answer and explanation
Correct option: B
x−tan⁡−1x1+x2+c\frac{x-\tan ^{-1} x}{\sqrt{1+x^2}}+c
Option C (tan⁡−1x−x1+x2+c\frac{\tan ^{-1} x-x}{\sqrt{1+x^2}}+c) is correct because substituting x=tan⁡θx = \tan \theta simplifies the integral, and integration by parts yields the correct result.
663 Easy
663 of 995
Application +4 | -1
If ∫tan⁡4x dx=Ktan⁡3x+Ltan⁡x+f(x)\int \tan ^4 x \mathrm{~d} x=K \tan ^3 x+L \tan x+f(x), then
View answer and explanation
Correct option: C
f(x)=x+Cf(x)=x+C
Option C (f(x)=x+Cf(x)=x+C) is correct because after integrating tan⁡4x\tan^4 x, the remaining term is x+Cx+C.
664 Easy
664 of 995
Analyzing +4 | -1
∫ex4(x+x3+2x5)ex2dx\int e^{x^4}\left(x+x^3+2 x^5\right) e^{x^2} d x is equal to
View answer and explanation
Correct option: D
12x2ex2ex4+C\frac{1}{2} x^2 e^{x^2} e^{x^4}+C
Option D (12x2ex2ex4+C\frac{1}{2} x^2 e^{x^2} e^{x^4}+C) is correct because by using the substitution x2=tx^2 = t and integrating by parts, the integral evaluates to 12x2ex2ex4+C\frac{1}{2} x^2 e^{x^2} e^{x^4}+C.
665 Easy
665 of 995
Understanding +4 | -1
∫xx(1+ln⁡x)dx\int x^x(1+\ln x) d x is equal to
View answer and explanation
Correct option: B
xx+cx^x+c
Option B (xx+cx^x+c) is correct because the integral of xx(1+ln⁡x)x^x(1+\ln x) is found by recognizing that it is the derivative of xxx^x.
666 Medium
666 of 995
Application +4 | -1
Integration of f(x)=1+x2f(x)=1+x^2 with respect to x2x^2, is
View answer and explanation
Correct option: B
23(1+x2)3/2+C\frac{2}{3}\left(1+x^2\right)^{3 / 2}+C
Option B (23(1+x2)3/2+C\frac{{2}}{{3}}(1+x^2)^{3/2}+C) is correct because it is the result of integrating the given function with respect to x2x^2.
667 Easy
667 of 995
Application +4 | -1
∫cos⁡2x(cos⁡x+sin⁡x)2dx=\int \frac{\cos 2 x}{(\cos x+\sin x)^2} d x=
View answer and explanation
Correct option: C
log⁡(cos⁡x+sin⁡x)+c\log (\cos x+\sin x)+c
Option C is correct because the integral simplifies to log⁡(cos⁡x+sin⁡x)+c\log(\cos x + \sin x) + c after using trigonometric identities and substitution.
668 Easy
668 of 995
Analyzing +4 | -1
The value of ∫ (x  +  1)2x(x2+  1) dx\int \,\frac{{{{\left( {x\,\, + \,\,1} \right)}^2}}}{{x\left( {{x^2} + \,\,1} \right)}}\,dx is equal to
View answer and explanation
Correct option: B
log⁡e  x+2  tan⁡−1  x  +  c{\log _e}\,\,x + 2\,\,{\tan ^{ - 1}}\,\,x\,\, + \,\,c
Option B (log⁡e x+2 tan⁡−1 x + c{\log _e}\,x + 2\,{\tan ^{ - 1}}\,x\, + \,c) is correct because the integrand can be split into simpler fractions, which integrate to this result.
669 Medium
669 of 995
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.
670 Medium
670 of 995
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.
671 Easy
671 of 995
Application +4 | -1
∫axxdx=\int \frac{a^{\sqrt{x}}}{\sqrt{x}} d x=
View answer and explanation
Correct option: B
2axlog⁡ae+c2 a^{\sqrt{x}} \log _a e+c
Option B (2axlog⁡ae+c2 a^{\sqrt{x}} \log _a e+c) is correct because it is the result of integrating the given expression using substitution and the properties of logarithms.
672 Easy
672 of 995
Application +4 | -1
The value of ∫xlog⁡x(log⁡x−1)dx\int x \log x(\log x-1) d x is equal to
View answer and explanation
Correct option: B
12(xlog⁡x−x)2+C\frac{1}{2}(x \log x-x)^2+C
Option B is correct because the integral simplifies to a form that can be directly integrated, resulting in 12(xlog⁡x−x)2+C\frac{1}{2}(x \log x-x)^2+C.
673 Easy
673 of 995
Analyzing +4 | -1
∫x2−2x4+4dx\int \frac{x^2-2}{x^4+4} d x is equal to
View answer and explanation
Correct option: C
14log⁡e(x2−2x+2x2+2x+2)+c\frac{1}{4} \log _e\left(\frac{x^2-2 x+2}{x^2+2 x+2}\right)+c
Option C is correct because the integral is solved by dividing by x2 and completing the square, leading to a logarithmic form.
674 Medium
674 of 995
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.
675 Easy
675 of 995
Application +4 | -1
If ∫dxx5x2−3=Ktan⁡−1f(x)+C\int \frac{d x}{x \sqrt{5 x^2-3}}=K \tan ^{-1} f(x)+C then
View answer and explanation
Correct option: B
f(x)=53x2−1,K=13f(x)=\sqrt{\frac{5}{3} x^2-1}, K=\frac{1}{\sqrt{3}}
Option B (f(x)=53x2−1,K=13f(x)=\sqrt{{\frac{{5}}{{3}} x^2-1}}, K=\frac{{1}}{{\sqrt{{3}}}}) is correct because the integral can be solved by substitution, leading to an arctangent integral.

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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
676 Medium
676 of 995
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.
677 Easy
677 of 995
Application +4 | -1
∫(1+x−x−1)ex+x−1dx\int\left(1+x-x^{-1}\right) e^{x+x^{-1}} d x, is equal to
View answer and explanation
Correct option: A
xex+x−1+Cx e^{x+x^{-1}}+C
Option A is correct because the integral simplifies to xex+x−1+Cx e^{x+x^{-1}}+C after applying integration by parts.
678 Easy
678 of 995
Analyzing +4 | -1
The value of ∫dxcos⁡6x+sin⁡6x\int \frac{d x}{\cos ^6 x+\sin ^6 x}, is equal to
View answer and explanation
Correct option: D
tan⁡−1(−2cot⁡2x)+C\tan ^{-1}(-2 \cot 2 x)+C
Option D (tan⁡−1(−2cot⁡2x)+C\tan^{-1}(-2 \cot 2x) + C) is correct because after simplifying the integral using trigonometric identities and substitution, we arrive at this result.
679 Easy
679 of 995
Application +4 | -1
The value of ∫1x2(x4+1)3/4dx\int \frac{1}{x^2\left(x^4+1\right)^{3 / 4}} d x, is
View answer and explanation
Correct option: B
−(1+1x4)1/4-\left(1+\frac{1}{x^4}\right)^{1 / 4}
Option B is correct because the integral simplifies to -(1 + 1/x4)(1/4) + C after substitution and simplification.
680 Easy
680 of 995
Application +4 | -1
∫x1+x4dx=\int \frac{x}{1+x^4} d x=
View answer and explanation
Correct option: B
12tan⁡−1x2+c\frac{1}{2} \tan ^{-1} x^2+c
Option B is correct because substituting t=x2t = x^2 transforms the integral to 12∫11+t2dt\frac{1}{2} \int \frac{1}{1 + t^2} dt, which equals 12tan⁡−1t+c=12tan⁡−1x2+c\frac{1}{2} \tan^{-1} t + c = \frac{1}{2} \tan^{-1} x^2 + c.
681 Medium
681 of 995
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.
682 Easy
682 of 995
Application +4 | -1
Let x2≠nπ−1,n∈Nx^2 \neq n \pi-1, n \in N. Then, the value of ∫x2sin⁡(x2+1)−sin⁡2(x2+1)2sin⁡(x2+1)+sin⁡2(x2+1)dx\int 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)}} d x is equal to
View answer and explanation
Correct option: B
log⁡∣sec⁡(x2+12)∣+C\log \left|\sec \left(\frac{x^2+1}{2}\right)\right|+C
Option B is correct because after simplifying the expression inside the square root and using a trigonometric identity, the integral can be evaluated as log⁡∣sec⁡(x2+12)∣+C\log \left|\sec \left(\frac{x^2+1}{2}\right)\right|+C.
683 Medium
683 of 995
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.
684 Medium
684 of 995
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.
685 Easy
685 of 995
Application +4 | -1
If ∫5+x10x16dx=a(1+5x10)3/2+C\int \frac{\sqrt{5+x^{10}}}{x^{16}} d x=a\left(1+\frac{5}{x^{10}}\right)^{3 / 2}+C, then a=a=
View answer and explanation
Correct option: C
−175-\frac{1}{75}
Option C is correct because after substitution and integration, the value of 'a' is found to be -1/75.
686 Easy
686 of 995
Application +4 | -1
∫dxa2cos⁡2x+b2sin⁡2x(a,b>0)\int \frac{\mathrm{d} x}{a^2 \cos ^2 x+b^2 \sin ^2 x}(a, b>0) is equal to
View answer and explanation
Correct option: C
1abtan⁡−1((ba)tan⁡x)+C\frac{1}{a b} \tan ^{-1}\left(\left(\frac{b}{a}\right) \tan x\right)+C
Option C (1abtan⁡−1((ba)tan⁡x)+C\frac{{1}}{{a b}} \tan ^{-1}\left(\left(\frac{{b}}{{a}}\right) \tan x\right)+C) is correct because dividing by cos⁡2x\cos^2 x and substituting t=tan⁡xt = \tan x leads to an arctangent integral.
687 Medium
687 of 995
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.
688 Easy
688 of 995
Application +4 | -1
If f(3x−43x+4)=x+2f\left(\frac{3 x-4}{3 x+4}\right)=x+2, then ∫f(x)dx\int f(x) d x is:
View answer and explanation
Correct option: B
−83log⁡∣1−x∣+23x+c-\frac{8}{3} \log |1-x|+\frac{2}{3} x+c
Option B is correct because the integral can be solved by substitution and partial fractions.
689 Easy
689 of 995
Application +4 | -1
If ∫f(x)x2−x+1 dx=32log⁡(x2−x+1)+\int \frac{f(x)}{x^2-x+1} \mathrm{~d} x=\frac{3}{2} \log \left(x^2-x+1\right)+ 13tan⁡−12x−13+C\frac{1}{\sqrt{3}} \tan ^{-1} \frac{2 x-1}{\sqrt{3}}+C then f(x)f(x) is equal to
View answer and explanation
Correct option: C
3x−13 x-1
The correct answer is 3x−13x-1 because differentiating the given integral and simplifying leads to f(x)=3x−1f(x) = 3x - 1.
690 Easy
690 of 995
Application +4 | -1
If ∫ex(1+x)sec⁡2(xex)dx=f(x)+\int e^x(1+x) \sec ^2\left(x e^x\right) d x=f(x)+ Constant, then f(x)f(x) is equal to
View answer and explanation
Correct option: D
tan⁡(xex)\tan \left(x e^x\right)
Option D (tan⁡(xex)\tan \left(x e^x\right)) is correct because the integral simplifies to tan⁡(xex)+C\tan(xe^x) + C, so f(x)=tan⁡(xex)f(x) = \tan(xe^x).
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