If x is any arbitrary constant, then $\smallint \,\,{2^{{2^{{2^x}}}}}\,\,.\,\,{2^{{2^x}}}\,\,.\,\,{2^x}\,\,dx$ is equal to

$\frac{{\smallint \,\,{2^2}^x\,\,dx}}{{{{\left( {\log \,\,2} \right)}^3}}}\,\, + \,\,c$

$\frac{{\smallint \,\,{2^{{2^2}}}^x\,\,dx}}{{{{\left( {\log \,\,2} \right)}^3}}}\,\, + \,\,c$

$\smallint \,\,{2^{{2^2}}}^x\,\,{\left( {\log \,\,2} \right)^3}\,\,dx\,\, + \,\,c$

$\smallint \,\,\sqrt {\frac{{1 + x}}{{1 - x}}} \,\,dx$ is equal to

$ - {\sin ^{ - 1}}\,\,x\,\, - \,\,\sqrt {{x^2}\,\, - \,\,1} + \,\,c$

The value of $\smallint \,{e^x}\,\,\frac{{1\,\, + \,\,n{x^{n - 1}}\,\, - \,\,{x^{2n}}}}{{\left( {1\,\, - \,\,{x^n}} \right)\,\,\sqrt {1\,\, - \,\,{x^{2n}}} }}\,\,dx$ is equal to

${e^x}\,\,\left( {\sqrt {1 - {x^2}} } \right)\,\, + \,\,c$

${e^x}\,\,\frac{{\sqrt {1 + {x^{2n}}} }}{{1 + {x^{2n}}}}\,\, + \,\,c$

$\frac{{{e^x}\,\sqrt {1 - {x^n}} }}{{1 - {x^{2n}}}}\,\, + \,\,c$

If $I=\int \frac{\sin 2 x}{(3+4 \cos x)^3} d x$, then $I=$

$\frac{3 \cos x+8}{(3+4 \cos x)^2}+C$

$\frac{3+\cos x}{(3+4 \cos x)^2}+C$

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JEE • MATHEMATICS • INDEFINITE INTEGRATION

Indefinite integration practice · Easy

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571 Easy

571 of 746

Analyzing +4 | -1

\int \frac{\cos 4 x+1}{\cot x-\tan x} d x=k \cos 4 x+c

then

View answer and explanation

Correct option: B

k=-1 / 8

Option B (

k=-1 / 8

) is correct because after simplifying the integral and solving, the value of k is found to be -1/8.

572 Easy

572 of 746

Analyzing +4 | -1

\int \,\,\log \,\,\left( {\sqrt {1 - x} \,\, + \,\,\sqrt {1 + x} } \right)dx = x\,\;f\left( x \right) + Ax + B\,\,{\sin ^{ - 1}}x + C

, then

View answer and explanation

Correct option: A

f\left( x \right) = \log \,\,\left( {\sqrt {1 - x} \,\, + \,\,\sqrt {1 + x} } \right)

Option A (

f\left( x \right) = \log \,\,\left( {\sqrt {1 - x} \,\, + \,\,\sqrt {1 + x} } \right)

) is correct because after integrating by parts, the function f(x) matches this logarithmic expression.

573 Easy

573 of 746

Understanding +4 | -1

\int \frac{\operatorname{cosec}^2 x}{1+\cot x} d x=

View answer and explanation

Correct option: B

-\log (1+\cot x)+c

Option B (

\log (1+\cot x)+c

) is correct because substituting

t = 1 + \cot x

leads to a simple logarithmic integral.

574 Easy

574 of 746

Application +4 | -1

\int e^x\left(1-\cot x+\cot ^2 x\right) d x=

View answer and explanation

Correct option: B

-e^x \cot x+C

Option B (

-e^x \cot x+C

) is correct because the integral simplifies to this form by recognizing the derivative of

-\cot x

1 + \cot^2 x = \csc^2 x

575 Easy

575 of 746

Application +4 | -1

\int \frac{1}{x \cos ^2(1+\log x)} d x=

View answer and explanation

Correct option: A

\tan (1+\log x)+c

Option A,

\tan (1+\log x)+c

, is the correct answer because it results from a direct substitution and integration of the given expression.

576 Easy

576 of 746

Analyzing +4 | -1

\int \sec ^{4 / 3} x \operatorname{cosec}^{8 / 3} x \,d x=a(\tan x)^{-5 / 3}+b(\tan x)^{1 / 3}+C,

then

5 a+b=

View answer and explanation

Correct option: C

Option C is correct because after simplifying and integrating, we find that

5a + b = 0

577 Easy

577 of 746

Application +4 | -1

\int \,\,{e^x}\,\,\left( {\frac{{1 - \,\,\sin \,\,x}}{{1\,\, - \,\,\cos \,\,x}}} \right)\,\,dx = f\left( x \right)\,\, + \,\,\mathrm{constant}

, then

f\left( x \right)

is equal to

View answer and explanation

Correct option: C

- \,{e^x}\,\,\cot \,\,\left( {\frac{x}{2}} \right)\,\, + \,\,c

Option C (

\ - \\,{e^x}\\\,\cot \\,\left( {\frac{x}{2}} \right)\\ + \\,c

) is correct because the integral is solved by using trigonometric identities to simplify the integrand and then recognizing a pattern for integration.

578 Easy

578 of 746

Application +4 | -1

The value of

I=\sqrt{2} \int \frac{\sin x}{\sin (x-\pi / 4)} d x

View answer and explanation

Correct option: C

x+\log |\sin (x-\pi / 4)|+C

Option C is correct because the integral can be solved by using a substitution and trigonometric identities.

579 Easy

579 of 746

Application +4 | -1

\int \frac{\log x d x}{x^3}=

View answer and explanation

Correct option: B

-\frac{1}{4 x^2}(2 \log x+1)+c

Option B (

-\frac{{1}}{{4 x^2}}(2 \log x+1)+c

) is correct because it is the result of integrating the given expression using integration by parts.

580 Easy

580 of 746

Application +4 | -1

\int\{\sin (\log x)+\cos (\log x)\} d x=

View answer and explanation

Correct option: A

x \sin (\log x)+c

Option A is correct because the integral of

\sin(\log x) + \cos(\log x)

simplifies to

x \sin(\log x) + c

after substitution and integration.

581 Easy

581 of 746

Analyzing +4 | -1

If x is any arbitrary constant, then

\int \,\,{2^{{2^{{2^x}}}}}\,\,.\,\,{2^{{2^x}}}\,\,.\,\,{2^x}\,\,dx

is equal to

View answer and explanation

Correct option: D

None of these

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

\frac{{{2^{{2^{{2^x}}}}}}}{{{{\left( {\log \,2} \right)}^3}}}\, + \,c

, which is not any of the given options.

582 Easy

582 of 746

Application +4 | -1

\int \,\,\sqrt {\frac{{1 + x}}{{1 - x}}} \,\,dx

is equal to

View answer and explanation

Correct option: C

{\sin ^{ - 1}}\,\,x\,\, - \,\,\sqrt {1\,\, - \,\,{x^2}} \, + \,\,c

Option C (

{\sin ^{ - 1}}\,x\, - \,\sqrt {1\, - \,{x^2}} \, + \,c

) is correct because after simplifying the integral and using substitution, we arrive at this result.

583 Easy

583 of 746

Application +4 | -1

\int \frac{e^{\tan ^{-1} x}}{1+x^2} d x=

View answer and explanation

Correct option: C

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

Option C is correct because the integral can be solved using a simple substitution.

584 Easy

584 of 746

Analyzing +4 | -1

\int \frac{2 x+5}{\sqrt{9 x^2+6 x+2}} d x =a \sqrt{9 x^2+6 x+2}+b \log _e\left(3 x+1+\sqrt{9 x^2+6 x+2}\right)+c

then

9(a+b)

is equal to

View answer and explanation

Correct option: C

Option C (41) is correct because after evaluating the integral and comparing coefficients, 9(a+b) equals 41.

585 Easy

585 of 746

Application +4 | -1

\int x^{-2 / 3}\left(1+x^{1 / 3}\right)^{1 / 2} d x

is equal to

View answer and explanation

Correct option: B

2\left(1+x^{1 / 3}\right)^{3 / 2}+c

Option B (

2(1+x^{1/3})^{3/2}+c

) is correct because it is the result of integrating the given expression using substitution.

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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

586 Easy

586 of 746

Analyzing +4 | -1

The value of

\int \,{e^x}\,\,\frac{{1\,\, + \,\,n{x^{n - 1}}\,\, - \,\,{x^{2n}}}}{{\left( {1\,\, - \,\,{x^n}} \right)\,\,\sqrt {1\,\, - \,\,{x^{2n}}} }}\,\,dx

is equal to

View answer and explanation

Correct option: D

\frac{{{e^x}\,\sqrt {1 - {x^{2n}}} }}{{1 - {x^n}}}\,\, + \,\,c

Option D (

\frac{{{e^x}\,\sqrt {1 - {x^{2n}}} }}{{1 - {x^n}}}\, + \,c

) is correct because it is the result of integrating the given expression by recognizing the form of

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

587 Easy

587 of 746

Application +4 | -1

\int\left(1-x^2\right) \log x d x=

View answer and explanation

Correct option: A

\left(x-\frac{x^3}{3}\right) \log x-\left(x-\frac{x^3}{9}\right)+c

Option A is correct because the integral of (1-x²)log(x) dx is found using integration by parts, resulting in the given expression.

588 Easy

588 of 746

Application +4 | -1

The primitive of the function

f(x)=x|\cos x|

, when

\frac{\pi}{2} < x < \pi

is given by

View answer and explanation

Correct option: B

-\cos x-x \sin x+C

Option B (

-\cos x-x \sin x+C

) is correct because it represents the antiderivative of the function

f(x)=x|\cos x|

in the given interval.

589 Easy

589 of 746

Application +4 | -1

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

View answer and explanation

Correct option: B

\tan \left(x e^x\right)

Option B (

\tan (x e^x)

) is correct because the integral simplifies to

\tan(xe^x) + c

after a simple substitution.

590 Easy

590 of 746

Application +4 | -1

I=\int \frac{\sin 2 x}{(3+4 \cos x)^3} d x

, then

I=

View answer and explanation

Correct option: B

\frac{3+8 \cos x}{16(3+4 \cos x)^2}+C

Option B is correct. The integral is solved using substitution, where

t = 3 + 4\cos x

, simplifying the expression and allowing for direct integration.

591 Easy

591 of 746

Application +4 | -1

Let

{I_n} = \int {{{\tan }^n}\,x\,dx\left( {n > 1} \right)}

. If

{I_4} + {I_6} = a\,{\tan ^5}x + b{x^5} + C

, where

C

is a constant of integration, then the ordered pair

(a,\,b)

is equal to

View answer and explanation

Correct option: B

\left( {\frac{1}{5},0} \right)

Option B

\left( {\frac{1}{5},0} \right)

is correct because by using the reduction formula for the integral of tan^n(x), we can find the values of a and b.

592 Easy

592 of 746

Understanding +4 | -1

\int\left[f(x) g^{\prime \prime}(x)-f^{\prime \prime}(x) g(x)\right] d x

is equal to

View answer and explanation

Correct option: C

f(x) g^{\prime}(x)-f^{\prime}(x) g(x)

Option C (

f(x) g^{\prime}(x)-f^{\prime}(x) g(x)

) is correct because it results from applying integration by parts twice to the given integral.

593 Easy

593 of 746

Application +4 | -1

\int \frac{\left(a^x-b^x\right)^2}{a^x b^x} d x

equals

View answer and explanation

Correct option: D

\frac{\left(\frac{a}{b}\right)^x-\left(\frac{b}{a}\right)^x}{\ln \left(\frac{a}{b}\right)}-2 x+c

Option D (

\frac{{(\frac{{a}}{{b}})^x-(\frac{{b}}{{a}})^x}}{{\ln (\frac{{a}}{{b}})}} - 2x + c

) is correct because it is derived by expanding the integrand and integrating each term.

594 Easy

594 of 746

Application +4 | -1

I=\int \frac{\log \left(x+\sqrt{1+x^2}\right)}{\sqrt{1+x^2}} d x

, then I equals

View answer and explanation

Correct option: B

(1 / 2)\left(\log \left(x+\sqrt{1+x^2}\right)\right)^2+C

Option B (\

1/2)(\log(x + \sqrt{1+x^2}))^2 + C

is correct. The integral is solved by using the substitution \

t = \log(x + \sqrt{1+x^2}))

595 Easy

595 of 746

Analyzing +4 | -1

\int \frac{f(x)}{x^3-1} d x

, where

f(x)

is a polynomial of degree 2 in

x

such that

f(0)=f(1)=3 f(2)=-3

and

\int \frac{f(x)}{x^3-1} d x=-\log |x-1|+\log \left|x^2+x+1\right|

+\frac{m}{\sqrt{n}} \tan ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right)+ C.

Then

(2 m+n)

View answer and explanation

Correct option: C

Option C (7) is correct because after solving the integral and comparing coefficients, we find m=2 and n=3, thus 2m+n=7.

596 Easy

596 of 746

Application +4 | -1

\int f(x) d x=\psi(x)

, then

\int x^5 f\left(x^3\right) d x

is equal to

View answer and explanation

Correct option: C

\frac{1}{3} x^3 \phi\left(x^3\right)-\int x^2 \phi\left(x^3\right) d x+C

Option C (

\frac{1}{3} x^3 \phi(x^3)-\int x^2 \phi(x^3) d x+C

) is correct because using substitution and integration by parts leads to the correct expression.

597 Easy

597 of 746

Application +4 | -1

\int \frac{2 x \tan ^{-1} x^2}{1+x^4} d x=

View answer and explanation

Correct option: B

\frac{1}{2}\left[\tan ^{-1} x^2\right]^2+c

Option B (

\frac{1}{2}\left[\tan ^{-1} x^2\right]^2+c

) is correct because it is the result of a u-substitution and applying the power rule for integration.

598 Easy

598 of 746

Application +4 | -1

The value of

\int x \sin k x d x

View answer and explanation

Correct option: D

-\frac{x \cos k x}{k}+\frac{\sin k x}{k^2}+c

Option D is correct because the integral of x sin(kx) dx is found using integration by parts, resulting in -x cos(kx)/k + sin(kx)/k² + c.

599 Easy

599 of 746

Application +4 | -1

The value of integral

\int \frac{\mathrm{d} x}{x \sqrt{1-x^3}}

is given by

View answer and explanation

Correct option: B

\frac{1}{3} \log \left|\frac{\sqrt{1-x^3}-1}{\sqrt{1-x^2}+1}\right|+C

Option B (

\frac{{1}}{{3}} \log |\frac{{\sqrt{{1-x^3}}-1}}{{\sqrt{{1-x^2}}+1}}| + C

) is correct because it is the result of using substitution and partial fractions to evaluate the integral.

600 Easy

600 of 746

Application +4 | -1

\int \limits_0^1 \,\,\frac{x}{{{{\left( {1 - x} \right)}^{3/4}}}}\,\,dx =

View answer and explanation

Correct option: B

\frac{{16}}{5}

Option B (

\frac{{16}}{5}

) is correct because after applying integration by parts and evaluating the definite integral, the result is 16/5.

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