Skip to main content
Back to INDEFINITE INTEGRATION

Your Progress

0 / 995 Answered
0
Correct
0
Incorrect
0%
Accuracy
JEE • MATHEMATICS • INDEFINITE INTEGRATION

Indefinite integration practice · Mixed difficulty

Choose an answer to check it, or open “View answer” to study the explanation. Sign in to save progress to your account.

Practice format and chapter quiz

These sets use single-correct-answer MCQs. They do not cover every JEE Main or JEE Advanced question format and are not full-length mock papers.

Take a chapter quiz →
Showing 451-480 of 995 questions

Preparing practice and saved progress…

451 Medium
451 of 995
Application +4 | -1
If ∫sin⁡  xsin⁡  (x−α)  dx=Ax+B  log⁡e  sin⁡ (x−α)+C\int \frac{{\sin \,\,x}}{{\sin \,\,\left( {x - \alpha } \right)}}\,\,dx = Ax + B\,\,{\log _e}\,\,\sin \,\left( {x - \alpha } \right) + C, then
View answer and explanation
Correct option: C
A=cos⁡  α,   B=sin⁡  αA = \cos \,\,\alpha ,\;\,B = \sin \,\,\alpha
Option C is correct because by expanding sin⁡x\sin x as sin⁡((x−α)+α)\sin((x-\alpha) + \alpha) and integrating, we find A=cos⁡αA = \cos \alpha and B=sin⁡αB = \sin \alpha.
452 Easy
452 of 995
Application +4 | -1
If ∫x3(1+x43)1/7dx=α(1+x43)β+c\int \sqrt[3]{x}\left(1+\sqrt[3]{x^4}\right)^{1 / 7} d x=\alpha\left(1+\sqrt[3]{x^4}\right)^\beta+c, then α+β−1\alpha+\beta^{-1} is
View answer and explanation
Correct option: D
4932\frac{49}{32}
Option D (4932\frac{49}{32}) is correct because after evaluating the integral and comparing coefficients, α+β−1\alpha+\beta^{-1} equals this value.
453 Medium
453 of 995
Application +4 | -1
∫x−6(1+2x3)2/3dx\int x^{-6}\left(1+2 x^3\right)^{2 / 3} d x equals
View answer and explanation
Correct option: A
−15(2+x−3)5/3+c\frac{-1}{5}\left(2+x^{-3}\right)^{5 / 3}+c
Option A (−15(2+x−3)5/3+c\frac{-1}{5}\left(2+x^{-3}\right)^{5 / 3}+c) is correct because after substitution and integration, the expression simplifies to this form.
454 Easy
454 of 995
Analyzing +4 | -1
∫dx(x  +  1)5  (x  + 2)34\int \frac{{dx}}{{\sqrt[4]{{{{\left( {x\,\, + \,\,1} \right)}^5}\,\,{{\left( {x\,\, + \,2} \right)}^3}}}}} is equal to
View answer and explanation
Correct option: C
− 4   (x  +  2x  +  1)1/4+  c- \,4\,\;{\left( {\frac{{x\,\, + \,\,2}}{{x\,\, + \,\,1}}} \right)^{1/4}} + \,\,c
Option C (− 4  (x + 2x + 1)1/4+ c- \,4\;{\left( {\frac{{x\, + \,2}}{{x\, + \,1}}} \right)^{1/4}} + \,c) is correct because it is the result of the integration after a suitable substitution.
455 Medium
455 of 995
Application +4 | -1
Let ∫  5  −  x2  +  x  dx\int \,\,\sqrt {\frac{{5\,\, - \,\,x}}{{2\,\, + \,\,x}}} \,\,dx equal
View answer and explanation
Correct option: B
x+2  5−x  +  7  sin⁡−1  x+27+C\sqrt {x + 2} \,\,\sqrt {5 - x} \,\, + \,\,7\,\,{\sin ^{ - 1}}\,\,\sqrt {\frac{{x + 2}}{7}} + C
Option B (x+2 5−x + 7 sin⁡−1 x+27 +C\sqrt {x + 2} \,\sqrt {5 - x} \, + \,7\,{\sin ^{ - 1}}\,\sqrt {\frac{{x + 2}}{7}} \, + C) is correct because after substituting 2+x=t22 + x = t^2, the integral can be solved using standard integral formulas.
456 Hard
456 of 995
Application +4 | -1
∫(xx)x(2xlog⁡ex+x)dx\int\left(x^x\right)^x\left(2 x \log _e x+x\right) d x is equal to
View answer and explanation
Correct option: B
(xx)x+C\left(x^x\right)^x+C
Option B is correct because the integral simplifies to the derivative of (x^x)^x, making the integration straightforward.
457 Easy
457 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 the integral can be solved by substitution and using the formula ∫axdx=axlog⁡ea+C=axlog⁡ae+C\int a^x dx = \frac{{a^x}}{{\log_e a}} + C = a^x \log_a e + C.
458 Medium
458 of 995
Application +4 | -1
The primitive function of the function f  (x)=  (a2−x2)x4f\,\,\left( x \right) = \,\,\frac{{\sqrt {\left( {{a^2} - {x^2}} \right)} }}{{{x^4}}} is
View answer and explanation
Correct option: C
c  −  (a2−x2)3/23a2 x3c\,\, - \,\,\frac{{{{\left( {{a^2} - {x^2}} \right)}^{3/2}}}}{{3{a^2}\,{x^3}}}
Option C (c,−,(a2−x2)3/23a2 x3c\\,\\ - \\,\frac{{{{\left( {{a^2} - {x^2}} \right)}^{3/2}}}}{{3{a^2}\,{x^3}}}) is correct because it is the result of integrating the given function using substitution.
459 Easy
459 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=
View answer and explanation
Correct option: D
xex+x−1+Cx e^{x+x^{-1}}+C
Option D (xex+x−1+Cx e^{x+x^{-1}}+C) is correct because integrating by parts simplifies the expression, leading to the given result.
460 Easy
460 of 995
Application +4 | -1
∫  log⁡  (x  +  1+x2)1  +  x2    dx\int \,\,\frac{{\log \,\,\left( {x\,\, + \,\,\sqrt {1 + {x^2}} } \right)}}{{\sqrt {1\,\, + \,\,{x^2}} }}\;\,\,dx is equal to
View answer and explanation
Correct option: D
12  [log⁡  (x  +  1  +  x2)]2+  c\frac{1}{2}\,\,{\left[ {\log \,\,\left( {x\,\, + \,\,\sqrt {1\,\, + \,\,{x^2}} } \right)} \right]^2} + \,\,c
Option D is correct because the integral is solved using substitution, resulting in 12 [log⁡ (x + 1 +x2)]2 + c\frac{1}{2}\,{\left[ {\log \,\left( {x\,+\,\sqrt {1\,+{x^2}} } \right)} \right]^2}\,+\,c.
461 Medium
461 of 995
Analyzing +4 | -1
∫x2−1x2+1⋅dx1+x4\int \frac{x^2-1}{x^2+1} \cdot \frac{d x}{\sqrt{1+x^4}} is equal to
View answer and explanation
Correct option: D
12Cos⁡−1x2x2+1+c\frac{1}{\sqrt{2}} \operatorname{Cos}^{-1} \frac{x \sqrt{2}}{x^2+1}+c
Option D is correct because the integral simplifies to an inverse cosine form after dividing by x2 and using a substitution.
462 Easy
462 of 995
Application +4 | -1
If I=∫cos⁡2x−cos⁡2αsin⁡x−sin⁡αdxI=\int \frac{\cos 2 x-\cos 2 \alpha}{\sin x-\sin \alpha} d x, then II equals
View answer and explanation
Correct option: B
2cos⁡x−2xsin⁡α+C2 \cos x-2 x \sin \alpha+C
Option B (2cos⁡x−2xsin⁡α+C2 \cos x-2 x \sin \alpha+C) is correct because after simplifying the integral using trigonometric identities, the result matches this option.
463 Easy
463 of 995
Application +4 | -1
∫01  x71  −  x4    dx\int \nolimits_0^1 \,\,\frac{{{x^7}}}{{\sqrt {1\,\, - \,\,{x^4}} }}\;\,\,dx is equal to
View answer and explanation
Correct option: B
13\frac{1}{3}
Option B (13\frac{1}{3}) is correct because it is the result of evaluating the definite integral using substitution and the Gamma function.
464 Medium
464 of 995
Application +4 | -1
∫xcos⁡x(xsin⁡x+cos⁡x)2dx\int \frac{x \cos x}{(x \sin x+\cos x)^2} d x is equal to
View answer and explanation
Correct option: B
−1(xsin⁡x+cos⁡x)+k-\frac{1}{(x \sin x+\cos x)}+k
Option B (−1(xsin⁡x+cos⁡x)+k\frac{-1}{(x \sin x+\cos x)}+k) is correct because the integral simplifies to this form after recognizing the derivative in the numerator.
465 Medium
465 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: C
x2[cos⁡(log⁡ex)+sin⁡(log⁡ex)]+c\frac{x}{2}\left[\cos \left(\log _e x\right)+\sin \left(\log _e x\right)\right]+c
Option C (x2[cos⁡(log⁡ex)+sin⁡(log⁡ex)]+c\frac{x}{2}\left[\cos \left(\log _e x\right)+\sin \left(\log _e x\right)\right]+c) is correct because by using integration by parts twice, we can solve the integral.

Explore Other Chapters in MATHEMATICS

View All →
SEQUENCE OF SERIES 1764 MCQs DEFINITE INTEGRATION 1443 MCQs MATRICES 1363 MCQs PROBABILITY 1146 MCQs COMPLEX NUMBERS 980 MCQs DIFFERENTIAL EQUATIONS 977 MCQs
466 Easy
466 of 995
Understanding +4 | -1
Let f(x)=1xln⁡(xex)f(x)=\frac{1}{x} \ln \left(\frac{x}{e^x}\right) then its primitive w.r.t. xx is
View answer and explanation
Correct option: C
12ln⁡2x−x+C\frac{1}{2} \ln ^2 \mathrm{x}-\mathrm{x}+\mathrm{C}
Option C (12ln⁡2x−x+C\frac{1}{2} \ln ^2 \mathrm{x}-\mathrm{x}+\mathrm{C}) is correct because it is the result of integrating the given function after simplification using logarithm properties.
467 Easy
467 of 995
Analyzing +4 | -1
The value of I=∫dx(a+bx2)b−ax2I=\int \frac{d x}{\left(a+b x^2\right) \sqrt{b-a x^2}}, is
View answer and explanation
Correct option: A
1a(a2+b2)tan⁡−1(xa2+b2ab−ax2)+C\frac{1}{\sqrt{a\left(a^2+b^2\right)}} \tan ^{-1}\left(\frac{x \sqrt{a^2+b^2}}{a \sqrt{b-a x^2}}\right)+C
Option A is correct because the integral evaluates to 1a(a2+b2)tan⁡−1(xa2+b2ab−ax2)+C\frac{1}{\sqrt{a(a^2+b^2)}} \tan ^{-1}\left(\frac{x \sqrt{a^2+b^2}}{a \sqrt{b-a x^2}}\right)+C after using a trigonometric substitution and simplification.
468 Medium
468 of 995
Application +4 | -1
∫  x4+1(1−x4)3/2  dx\int \,\,\frac{{{x^4} + 1}}{{{{\left( {1 - {x^4}} \right)}^{3/2}}}}\,\,dx is equal to
View answer and explanation
Correct option: A
x1  −  x4  +  C\frac{x}{{\sqrt {1\,\, - \,\,{x^4}} }}\,\, + \,\,C
Option A (x1  −  x4  +  C\frac{x}{{\sqrt {1\,\, - \,\,{x^4}} }}\,\, + \,\,C) is correct because the integral can be solved by rewriting the expression and using a suitable substitution.
469 Medium
469 of 995
Application +4 | -1
∫  ∣ x ∣  log⁡  ∣ x ∣  dx\int \,\,\left| {\,x\,} \right|\,\,\log \,\,\left| {\,x\,} \right|\,\,dx equals (x≠0)\left( {x \ne 0} \right)
View answer and explanation
Correct option: D
12x  ∣ x ∣  log⁡   ∣ x ∣  −  14  x  ∣ x ∣  +  c\frac{1}{2}x\,\,\left| {\,x\,} \right|\,\,\log \,\,\,\left| {\,x\,} \right|\,\, - \,\,\frac{1}{4}\,\,x\,\,\left| {\,x\,} \right|\,\, + \,\,c
Option D (\frac{1}{2}x\,\left| {\,x\} \right|\,\log \,\,\left| {\,x\} \right|\, - \,\frac{1}{4}\,x\,\left| {\,x\} \right|\, + \,c\frac{1}{2}x\,\left| {\,x\} \right|\,\log \,\,\left| {\,x\} \right|\, - \,\frac{1}{4}\,x\,\left| {\,x\} \right|\, + \,c) is correct because it is the result of integrating |x|log|x| using integration by parts, considering both positive and negative cases for x.
470 Easy
470 of 995
Application +4 | -1
∫sec⁡xlog⁡(sec⁡x+tan⁡x)dx=\int \sec x \log (\sec x+\tan x) d x=
View answer and explanation
Correct option: B
12[log⁡(sec⁡x+tan⁡x)]2+c\frac{1}{2}[\log (\sec x+\tan x)]^2+c
Option B is correct because the integral evaluates to 12[log⁡(sec⁡x+tan⁡x)]2+c\frac{1}{2}[\log (\sec x+\tan x)]^2+c after a simple substitution.
471 Medium
471 of 995
Application +4 | -1
∫x2−1(x2+1)x4+1dx=\int \frac{x^2-1}{\left(x^2+1\right) \sqrt{x^4+1}} d x=
View answer and explanation
Correct option: B
12sec⁡−1(x2+12x)+c\frac{1}{\sqrt{2}} \sec ^{-1}\left(\frac{x^2+1}{\sqrt{2} x}\right)+c
Option B (12sec⁡−1(x2+12x)+c\frac{1}{\sqrt{2}} \sec ^{-1}\left(\frac{x^2+1}{\sqrt{2} x}\right)+c) is correct because it is the result of the integration after appropriate substitution and simplification.
472 Easy
472 of 995
Analyzing +4 | -1
If ∫cos⁡2xsin⁡4xcos⁡4x(1+cos⁡22x)dx=2log⁡e(1+cos⁡2x)−log⁡e(1+cos⁡22x)+f(x)+c\int \frac{\cos 2 x \sin 4 x}{\cos ^4 x\left(1+\cos ^2 2 x\right)} d x=2 \log _e(1+\cos 2 x)-\log _e\left(1+\cos ^2 2 x\right)+f(x)+\mathrm{c} then f(x)f(x) is
View answer and explanation
Correct option: A
sec⁡2x\sec ^2 x
Option A is correct because the integral can be solved by trigonometric manipulation and partial fractions, leading to the given expression for f(x).
473 Medium
473 of 995
Application +4 | -1
∫xa3−x3dx=\int \frac{\sqrt{x}}{\sqrt{a^3-x^3}} d x=
View answer and explanation
Correct option: B
23Sin⁡−1(xa)3/2+c\frac{2}{3} \operatorname{Sin}^{-1}\left(\frac{x}{a}\right)^{3 / 2}+c
Option B (23Sin⁡−1(xa)3/2+c\frac{2}{3} \operatorname{Sin}^{-1}\left(\frac{x}{a}\right)^{3 / 2}+c) is correct because by substituting x=asin⁡2/3θx=a \sin ^{2 / 3} \theta, we can evaluate the integral.
474 Easy
474 of 995
Application +4 | -1
∫32x3(log⁡x)2dx\int 32 x^3(\log x)^2 d x is equal to
View answer and explanation
Correct option: A
x4{8(log⁡x)2−4(log⁡x)+1}+cx^4\left\{8(\log x)^2-4(\log x)+1\right\}+c
Option A (x4{8(log⁡x)2−4(log⁡x)+1}+cx^4\left\{8(\log x)^2-4(\log x)+1\right\}+c) is correct because it is the result of integrating the given expression using integration by parts twice.
475 Easy
475 of 995
Application +4 | -1
∫(x+1)(x+log⁡x)2xdx=\int \frac{(x+1)(x+\log x)^2}{x} d x=
View answer and explanation
Correct option: C
13(x+log⁡x)3+c\frac{1}{3}(x+\log x)^3+c
Option C is correct because the integral simplifies to 13(x+log⁡x)3+c\frac{1}{3}(x + \log x)^3 + c after applying the substitution method.
476 Easy
476 of 995
Application +4 | -1
∫  3x9x  −  1    dx\int \,\,\frac{{{3^x}}}{{\sqrt {{9^x}\,\, - \,\,1} }}\,\,\;dx is equal to
View answer and explanation
Correct option: A
1log⁡  3  log⁡  ∣ 3x  +  9x−1 ∣  +  c\frac{1}{{\log \,\,3}}\,\,\log \,\,\left| {\,{3^x}\,\, + \,\,\sqrt {{9^x} - 1\,} } \right|\,\, + \,\,c
Option A is correct because the integral can be solved by substituting 3x=t{3^x} = t and using the standard integral formula.
477 Medium
477 of 995
Application +4 | -1
∫1x2+2 d (x2+1)\int \frac{1}{\sqrt{x^2+2}} \,d\,\left(x^2+1\right) is equal to
View answer and explanation
Correct option: A
2x2+2+C2 \sqrt{x^2+2}+C
Option A (2x2+2+C2 \sqrt{{x^2+2}}+C) is correct because it is the direct result of integrating the given expression.
478 Medium
478 of 995
Analyzing +4 | -1
If ∫dx(x2+a2)3=x4a2(x2+a2)+mna2\int \frac{d x}{\left(x^2+a^2\right)^3}=\frac{x}{4 a^2\left(x^2+a^2\right)}+\frac{m}{n a^2} {x2a2(x2+a2)+12a3tan⁡−1(xa)}+C\left\{\frac{x}{2 a^2\left(x^2+a^2\right)}+\frac{1}{2 a^3} \tan ^{-1}\left(\frac{x}{a}\right)\right\}+C. Then ∣m−n∣|m-n| is equal to
View answer and explanation
Correct option: D
1
Option D is correct because after evaluating the integral and comparing coefficients, the absolute difference between m and n is found to be 1.
479 Easy
479 of 995
Application +4 | -1
∫ax1−a2xdx=\int \frac{a^x}{\sqrt{1-a^{2 x}}} d x=
View answer and explanation
Correct option: A
1log⁡asin⁡−1ax+c\frac{1}{\log a} \sin ^{-1} a^x+c
Option A is correct because the integral can be solved using substitution.
480 Easy
480 of 995
Application +4 | -1
∫  ex  (x−1)x2    dx\int \,\,{e^x}\,\,\frac{{\left( {x - 1} \right)}}{{{x^2}}}\;\,\,dx is equal to
View answer and explanation
Correct option: C
exx  +  c\frac{{{e^x}}}{x}\,\, + \,\,c
Option C (exx + c\frac{{{e^x}}}{x}\, + \,c) is correct because the integral can be simplified and solved using the property ∫ex[f(x)+f′(x)]dx=exf(x)+c\int e^x [f(x) + f'(x)] dx = e^x f(x) + c.
Revision notes and key takeaways

Review INDEFINITE INTEGRATION in your textbook or class notes, then use this set to check your understanding.

Showing 451-480 of 995 questions

Continue Your Practice

📚 More Sets in This Chapter

Try different difficulty levels to master this topic

→ ENGINEERING MATHEMATICS - INDEFINITE INTEGRATION MCQs (Easy) 746 MCQs → ENGINEERING MATHEMATICS - INDEFINITE INTEGRATION MCQs (Hard) 15 MCQs → ENGINEERING MATHEMATICS - INDEFINITE INTEGRATION MCQs (Medium) 411 MCQs
🎯 Next Chapters to Practice

Continue building your subject mastery

→ SEQUENCE OF SERIES 1764 MCQs → DEFINITE INTEGRATION 1443 MCQs → MATRICES 1363 MCQs → PROBABILITY 1146 MCQs

Your practice

0 / 995 answered · 995 remaining · 0% accuracy

Progress is stored on this device. Sign in to save to your account.

✓ Correct · ✗ Incorrect · ○ Unanswered

Practice settings

Type at least 2 characters or jump directly:
🩺 NEET MCQs 🔬 JEE MCQs 🎯 JEE College Predictor Study guides 📊 Dashboard
🔓

Save your learning progress

Sign in with a free account to save your practice answers and quiz results, then return to them when you revise.

Continue with Free Account →