$\smallint \,\,\sqrt {\frac{{\cos \,\,x\,\, - \,\,{{\cos }^3}\,x}}{{1\,\, - \,\,{{\cos }^3}\,\,x}}} \,\,dx$ is equal to

$\frac{2}{3}\,\,{\sin ^{ - 1}}\left( {{{\cos }^{3/2}}\,\,x} \right) + C$

$\frac{3}{2}\,\,{\sin ^{ - 1}}\left( {{{\cos }^{3/2}}\,\,x} \right) + C$

If $\int \tan ^7 x d x=f(x)+\ln |\cos x|+c$, then

$f(x)$ is a polynomial of degree 8 in $\tan x$

$f(x)$ is a polynomial of degree 5 in $\tan x$

$f(x)=\tan ^6 x-\frac{1}{4} \tan ^4 x+\frac{1}{2} \tan ^2 x+\ln |\cos x|+c$

$\smallint \,\,\frac{{dx}}{{{{\left( {{a^2} + {x^2}} \right)}^{3/2}}}}$ is equal to

$\frac{x}{{{{\left( {{a^2} + {x^2}} \right)}^{1/2}}}}\,\, + \,\,c$

$\frac{1}{{{a^2}{{\left( {{a^2} + {x^2}} \right)}^{1/2}}}} + c$

$\smallint \frac{{dx}}{{\cos \,\,x\,\, - \,\,\sin \,\,x}}$ is equal to

$\frac{1}{{\sqrt 2 }}\,\,\log \,\,\left| {\,\tan \,\,\left( {\frac{x}{2}\,\, - \,\,\frac{\pi }{8}} \right)} \right| + c$

$\frac{1}{{\sqrt 2 }}\,\,\log \,\,\left| {\,\cot \,\,\left( {\frac{x}{2}} \right)\,} \right|\, + c$

$\frac{1}{{\sqrt 2 }}\,\,\log \,\,\left| {\,\tan \,\,\left( {\frac{x}{2}\,\, - \,\,\frac{{3\pi }}{8}} \right)\,} \right| + c$

Back to INDEFINITE INTEGRATION

Your Progress

0 / 411 Answered

Correct

Incorrect

Accuracy

JEE • MATHEMATICS • INDEFINITE INTEGRATION

Indefinite integration practice · Medium

Try a question, then open “View answer” to check it. Sign in to submit answers and save progress.

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 391-411 of 411 questions

391 Medium

391 of 411

Application +4 | -1

The value of

\int \frac{x^7}{\left(1-x^2\right)^5} d x

, is

View answer and explanation

Correct option: B

\frac{1}{8} \frac{x^8}{\left(1-x^2\right)^4}+C

Option D (None of these) is correct because the integral evaluates to

\frac{1}{8} \frac{x^8}{(1-x^2)^4}+C

, but the explanation provided uses a substitution that changes the variable to theta and then back to x, which is not necessary.

392 Medium

392 of 411

Understanding +4 | -1

f_n(x)=\log \log \ldots \log x

(log is repeated

n

times) then

\int\left[x f_1(x) f_2(x) \ldots f_n(x)\right]^{-1} \mathrm{~d} x

is equal to

View answer and explanation

Correct option: A

f_{n+1}(x)+C

Option A (

f_{n+1}(x)+C

) is correct because it is the result of applying the substitution method and integrating.

393 Medium

393 of 411

Application +4 | -1

\int {\frac{{{x^2} + {{\sin }^2}x}}{{1 + {x^2}}}{{\sec }^2}xdx}

is equal to

View answer and explanation

Correct option: B

\tan x - {\tan ^{ - 1}}x + c

Option B (

\frac{\left(x-\frac{1}{x}\right)^{K+8}}{k+8}+c

) is correct because after simplifying and substituting

t = x - \frac{1}{x}

, the integral becomes a simple power rule integral.

394 Medium

394 of 411

Analyzing +4 | -1

\int e^{\sec x} \cdot \sec ^3 x \left(\sin ^2 x+\cos x+\sin x+\sin x \cos x\right) d x=

View answer and explanation

Correct option: A

e^{\sec x}(\sec x+\tan x)+

The correct answer is A (

e^{\sec x}(\sec x+\tan x)+c

) because the integral simplifies to this expression after applying integration by parts and simplification.

395 Medium

395 of 411

Application +4 | -1

\int x \tan ^{-1} x d x=

View answer and explanation

Correct option: A

\frac{1}{2}\left(x^2+1\right) \tan ^{-1} x-\frac{1}{2} x+c

Option A is correct because the integral is solved using integration by parts, simplifying the resulting expression to match the given form.

396 Medium

396 of 411

Analyzing +4 | -1

\int \,\,\sqrt {\frac{{\cos \,\,x\,\, - \,\,{{\cos }^3}\,x}}{{1\,\, - \,\,{{\cos }^3}\,\,x}}} \,\,dx

is equal to

View answer and explanation

Correct option: C

\frac{2}{3}\,\,{\cos ^{ - 1}}\,\,\left( {{{\cos }^{3/2}}\,\,x} \right) + C

Option C (

\frac{2}{3}\\,{\cos ^{ - 1}}\\,\left( {{{{

\cos

}}^{3/2}}\\,x} \right) + C

) is correct because it represents the result of the integration after simplification and substitution.

397 Medium

397 of 411

Application +4 | -1

\int \tan ^7 x d x=f(x)+\ln |\cos x|+c

, then

View answer and explanation

Correct option: D

f(x)

is a polynomial of degree 6 in

\tan x

Option D (

f(x)

is a polynomial of degree 6 in

\tan x

) is correct because the integral of

\tan^7 x

can be expressed as a polynomial of degree 6 in

\tan x

plus

\ln |\cos x| + c

398 Medium

398 of 411

Application +4 | -1

\int e^x \frac{(1-x)^2}{\left(1+x^2\right)^2} d x=

View answer and explanation

Correct option: C

e^x\left(\frac{1}{1+x^2}\right)+c

Option C (

e^x\left(\frac{1}{1+x^2}\right)+c

) is correct because the integral is solved using the form

\int e^x [f(x) + f'(x)] dx = e^x f(x) + c

399 Medium

399 of 411

Application +4 | -1

The value of

\int x{\left( {{x^x}} \right)^x}\,\,\left( {2\,\,\log \,\,x\,\, + \,\,1} \right)\,\,dx

View answer and explanation

Correct option: A

{\left( {{x^x}} \right)^x}\,\, + \,\,c

Option A (

{\left( {{x^x}} \right)^x}\\ + \\ c

) is correct because the integral is solved by recognizing the derivative of (x^x)^x and using substitution.

400 Medium

400 of 411

Analyzing +4 | -1

Integration of

\frac{1}{{\sqrt {{x^2} + \,\,9} }}

with respect to

\left( {{x^2} + \,\,1} \right)

is equal to

View answer and explanation

Correct option: C

2\,\sqrt {{x^2} + \,\,9} \,\, + \,\,C

The correct answer is C (

2\\,\sqrt {{{{x^2} + \,\,9}}} \,\, + \,\,C

) because the integral is solved by recognizing the differential and integrating.

401 Medium

401 of 411

Analyzing +4 | -1

\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

\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 (

\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.

402 Medium

402 of 411

Application +4 | -1

Let

\int e^x\left(f(x)-f^{\prime}(x)\right) d x=\phi(x)

, then

\int e^x f(x) d x

is equal to

View answer and explanation

Correct option: C

\frac{1}{2}\left\{\phi(x)+e^x t(x)\right\}+c

Option C (

\frac{1}{2}\{\phi(x)+e^x t(x)\}

+c) is correct because by integrating

e^x(f(x)+f'(x))

and

e^x(f(x)-f'(x))

and adding them, we can isolate the integral of

e^x f(x)

403 Medium

403 of 411

Analyzing +4 | -1

Given,

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

\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.

404 Medium

404 of 411

Application +4 | -1

Evaluate:

\int \frac{\sin (x-a)}{\sin (x+a)} d x

View answer and explanation

Correct option: C

(x+a) \cos 2 a-\sin 2 a \log |\sin (x+a)|+c

Option 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=t

and using trigonometric identities.

405 Medium

405 of 411

Application +4 | -1

\int \,\,\frac{{dx}}{{{{\left( {{a^2} + {x^2}} \right)}^{3/2}}}}

is equal to

View answer and explanation

Correct option: B

\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

\frac{x}{{{{a^2}{{\left( {{a^2} + {x^2}} \right)}^{1/2}}}}} + \,c

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

406 Medium

406 of 411

Application +4 | -1

\int \,\,\frac{{\sqrt x }}{{1\,\, + \,\,\sqrt[4]{{{x^3}}}}}\,\,dx

is equal to

View answer and explanation

Correct option: B

\frac{4}{3}\,\,\left[ {1 + {x^{3/4}}\,\, - \,\,{{\log }_e}\,\,\left( {1\,\, + \,\,{x^{3/4}}} \right)} \right] + C

Option B (

\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.

407 Medium

407 of 411

Understanding +4 | -1

n

is an odd positive integer, then

\int\left|x^n\right| d x

is equal to

View answer and explanation

Correct option: C

\frac{|x|^n x}{n+1}+C

Option C (

\frac{|x|^n x}{n+1}+C

) is correct because it accounts for both positive and negative values of

x

when

n

is an odd positive integer.

408 Medium

408 of 411

Application +4 | -1

\int \frac{d x}{x-x^3}=A \ln \left|\frac{x^2}{1-x^2}\right|+c

, then

A

is equal to

View answer and explanation

Correct option: B

1/2

Option B (1/2) is correct because the integral simplifies to the given form with A = 1/2 after partial fraction decomposition.

409 Medium

409 of 411

Application +4 | -1

\int \frac{d x}{x \log x \log (\log x)}=

View answer and explanation

Correct option: B

\log [\log (\log x)]+c

Option B (

\log [\log (\log x)]+c

) is correct because it is the result of two u-substitutions.

410 Medium

410 of 411

Application +4 | -1

\int \frac{{dx}}{{\cos \,\,x\,\, - \,\,\sin \,\,x}}

is equal to

View answer and explanation

Correct option: D

\frac{1}{{\sqrt 2 }}\,\,\log \,\,\left| {\,\tan \,\,\left( {\frac{x}{2}\,\, + \,\,\frac{{3\pi }}{8}} \right) + c\,} \right|

Option D is correct. The integral

\int \frac{{dx}}{{\cos \,\,x\,\, - \,\,\sin \,\,x}}

can be solved by rewriting the denominator and using trigonometric identities.

411 Medium

411 of 411

Analyzing +4 | -1

\int \frac{\tan x}{1+\tan x+\tan ^2 x} d x=x-\frac{K}{\sqrt{A}} \tan ^{-1}\left(\frac{K \tan x+1}{\sqrt{A}}\right)+C,

(

C

is a constant of integration), then the ordered pair

(K, A)

is equal to

View answer and explanation

Correct option: A

(2,3)

Option A ((2,3)) is correct because after solving the integral and comparing with the given form, we find K=2 and A=3.

Revision notes and key takeaways

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

Showing 391-411 of 411 questions

Continue Your Practice

📚 Switch Set (MEDIUM)

Chapter Sets

ALL 995 MCQs EASY 746 MCQs HARD 15 MCQs MEDIUM 411 MCQs

0 /411

0✓ 0✗

0% Accuracy