$\smallint \,\,\left[ {{e^{a\,\,\log \,\,x}}\,\, + \,\,{e^{x\,\,\log \,\,a}}} \right]\,\,dx\,\, = ... + \,\,c\,\,;\,\,a,\,\,x\,\, > 1$

$\frac{{{e^{a\,\,\log \,x}}}}{{a\,\,\log \,\,x}}\,\, + \,\,\frac{{{e^{x\,\,\log \,\,a}}}}{{x\,\,\log \,\,a}}$

$\frac{{{x^{a - 1}}}}{{a - 1}}\,\, + \,\,{a^x}\,\,.\,\,\log \,\,a$

$\frac{{{e^{a\,\,\log \,\,x}}}}{{a/x}}\,\, + \,\,\frac{{{e^{x\,\,\log \,\,a}}}}{x}$

If ${I_1} = \smallint {\sin ^{ - 1}}x\,dx$ and ${I_2} = \smallint {\sin ^{ - 1}}\,\,\sqrt {1\,\, - \,\,{x^2}} \;dx$, then

${I_2} = \frac{\pi }{2}\,\,{I_1}$

If $\int f(x) d x=F(x)$, then $\int x^3 f\left(x^2\right) d x$ is equal to

$\frac{1}{2}\left[x^2\{F(x)\}^2-\int\{F(x)\}^2 d x\right]$

$\frac{1}{2}\left[x^2 F(x)-\frac{1}{2} \int\{F(x)\}^2 d x\right]$

$\smallint \,\,\frac{1}{{x\left( {{x^n} + 1} \right)}}\,\,dx$ is equal to

$\frac{1}{n}\,\,\log \,\,\left( {\frac{{{x^n} + 1}}{{{x^n}}}} \right)$

$\smallint \,\,{e^x}\left( {\log \,\,\sin \,\,x\,\, + \,\,\cot \,\,x} \right)\,\,dx$ is equal to

${e^x}\,\,\log \,\,\sin \,\,x + \,\,\tan \,\,x + \,\,c$

$\smallint \,\,\left( {1 + x - {x^{ - 1}}} \right){e^{x + {x^{ - 1}}}}\,\,dx$ is equal to

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

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

$\smallint \,\,\frac{{dx}}{{\sin \,\,x - \cos \,\,x + \sqrt 2 }}$ equals

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

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

If $\int f(x) d x=F(x)$, then $\int x^3 f\left(x^2\right) d x$ is equal to

$\frac{1}{2}\left[x^2\{F(x)\}^2-\int\{F(x)\}^2 d x\right]$

$\frac{1}{2}\left[x^2 F(x)-\frac{1}{2} \int\{F(x)\}^2 d x\right]$

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JEE • MATHEMATICS • INDEFINITE INTEGRATION

Indefinite integration practice · Mixed difficulty

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Showing 631-660 of 995 questions

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631 Medium

631 of 995

Application +4 | -1

\int \frac{3 x^2+2 x}{\sqrt{x^6+2 x^5+x^4-2 x^3-2 x^2+5}} d x

equals

View answer and explanation

Correct option: A

\sinh ^{-1}\left(\frac{x^3+x^2-1}{2}\right)+c

Option A (

\sinh ^{-1}\left(\frac{x^3+x^2-1}{2}\right)+c

) is correct because it results from completing the square and using the inverse hyperbolic sine integral.

632 Easy

632 of 995

Understanding +4 | -1

Let

f

and

g

be two polynomials then

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

is equal to (ignoring the constant of integration)

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 is the result of applying integration by parts twice to the given integral.

633 Easy

633 of 995

Application +4 | -1

\int \,\,\left[ {{e^{a\,\,\log \,\,x}}\,\, + \,\,{e^{x\,\,\log \,\,a}}} \right]\,\,dx\,\, = ... + \,\,c\,\,;\,\,a,\,\,x\,\, > 1

View answer and explanation

Correct option: A

\frac{{{x^{a + 1}}}}{{a + 1}}\,\, + \,\,\frac{{{a^x}}}{{\log \,\,a}}

Option A is correct because the integral of the given expression simplifies to the sum of a power function and an exponential function, which matches option A.

634 Easy

634 of 995

Analyzing +4 | -1

{I_1} = \int {\sin ^{ - 1}}x\,dx

and

{I_2} = \int {\sin ^{ - 1}}\,\,\sqrt {1\,\, - \,\,{x^2}} \;dx

, then

View answer and explanation

Correct option: C

{I_1} + \,\,{I_2} = \frac{\pi }{2}\,\,x

Option C (

{I_1} + \, {I_2} = \frac{\pi }{2}\,x

) is correct because the sum of the integrals simplifies to an integral involving the constant

\frac{\pi}{2}

635 Easy

635 of 995

Application +4 | -1

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

, then

\int x^3 f\left(x^2\right) d x

is equal to

View answer and explanation

Correct option: B

\frac{1}{2}\left[x^2 F\left(x^2\right)-\int F\left(x^2\right) d\left(x^2\right)\right]

Option B (

\frac{1}{2}[x^2 F(x^2)-\int F(x^2) d(x^2)]

) is correct because it is the result of applying integration by parts to the integral

\int x^3 f(x^2) dx

with a suitable substitution.

636 Easy

636 of 995

Analyzing +4 | -1

\int \frac{x^2-1}{x \sqrt{x^4+3 x^2+1}} d x

is equal to

View answer and explanation

Correct option: A

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

Option A (

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

) is correct because after dividing by

x^2

and completing the square, the integral can be solved using a standard logarithmic form.

637 Medium

637 of 995

Analyzing +4 | -1

\int e^x \frac{1+10 x^9-x^{20}}{\left(1-x^{10}\right) \sqrt{1-x^{20}}} d x

is equal to: (where

c

is an integral constant)

View answer and explanation

Correct option: C

\quad e^x\left(\sqrt{\frac{1+x^{10}}{1-x^{10}}}\right)+c

Option C (

e^x\left(\sqrt{{\frac{{1+x^{{10}}}}{{1-x^{{10}}}}}}\right)+c

) is correct because the integral can be expressed in the form

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

638 Easy

638 of 995

Application +4 | -1

\int e^x \sin x d x=

View answer and explanation

Correct option: B

\frac{1}{2} e^x(\sin x-\cos x)+c

Option B (

\frac{1}{2} e^x(\sin x-\cos x)+c

) is correct because it is the result of applying integration by parts twice to the integral of

e^x \sin x

639 Easy

639 of 995

Analyzing +4 | -1

The integral

\int \frac{2 x^{12}+5 x^9}{\left(x^5+x^3+1\right)^3} d x

is equal to
where,

C

is an arbitrary constant.

View answer and explanation

Correct option: B

\frac{x^{10}}{2\left(x^5+x^3+1\right)^2}+C

Option B is correct because after substitution and simplification, the integral evaluates to the given expression.

640 Easy

640 of 995

Application +4 | -1

The value of

\int \frac{\sqrt{1+x}}{x} d x

, is

View answer and explanation

Correct option: A

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

Option A is correct because the integral is solved using a substitution method, resulting in the given expression with a logarithmic term.

641 Easy

641 of 995

Application +4 | -1

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

is equal to

View answer and explanation

Correct option: A

\frac{1}{n}\,\,\log \,\,\left( {\frac{{{x^n}}}{{{x^n} + 1}}} \right)\,\, + \,\,C

Option A (

\frac{1}{n}\,\,\log \,\,\left( {\frac{{{x^n}}}{{{x^n} + 1}}} \right)\, + \,C

) is correct because the integral is solved using substitution and partial fractions.

642 Easy

642 of 995

Understanding +4 | -1

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

View answer and explanation

Correct option: C

\frac{e^x}{x}+c

Option C (

\frac{{e^x}}{{x}}+c

) is correct because the integral is of the form

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

, where

f(x) = \frac{{1}}{{x}}

and

f'(x) = -\frac{{1}}{{x^2}}

643 Easy

643 of 995

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 is correct because the integral simplifies to

\tan(1 + \log x) + c

after applying the substitution method.

644 Easy

644 of 995

Application +4 | -1

\int \nolimits_0^{10} \,\,\left| {\,x\,\,\left( {x - 1} \right)\,\,\left( {x - 2} \right)} \right|\,\,dx\,\,\mathrm{is}\,\mathrm{\;equal\;}\,\mathrm{to}

is equal to

View answer and explanation

Correct option: B

1600.5

The correct answer is A (1600.5) because the integral is evaluated by splitting the integral into intervals where the function inside the absolute value is either positive or negative.

645 Easy

645 of 995

Application +4 | -1

\int \,\,\frac{{dx}}{{5 + 4\,\,\cos \,\,x}} = A\,\,{\tan ^{ - 1}}\left( {B\,\,\tan \,\,x/2} \right)\,\, + \,\,C

, then

View answer and explanation

Correct option: B

A = 2/3,\;\,B = 1/3

Option B (

A = 2/3,\;\,B = 1/3

) is correct because after applying trigonometric substitutions and integrating, the values of A and B match the given form.

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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

646 Easy

646 of 995

Application +4 | -1

\int \,\,{e^x}\left( {\log \,\,\sin \,\,x\,\, + \,\,\cot \,\,x} \right)\,\,dx

is equal to

View answer and explanation

Correct option: B

{e^x}\,\,\log \,\,\sin \,\,x\,\, + \,\,c

Option B (

{e^x}\,\,\log \,\,\sin \,\,x\,\, + \,\,c

) is correct because the integral matches the form

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

, where

f(x) = \log(\sin x)

647 Medium

647 of 995

Application +4 | -1

\int \frac{x-\sin x}{1-\cos x} d x=

View answer and explanation

Correct option: B

-x \cot \frac{x}{2}+c

Option B (

-x \cot \frac{x}{2}+c

) is correct because it results from integrating the given expression after using trigonometric identities and integration by parts.

648 Easy

648 of 995

Understanding +4 | -1

\int \frac{10 x^9+10^x \log _e 10}{10^x+x^{10}} d x=

View answer and explanation

Correct option: B

\log \left(10^x+x^{10}\right)+c

Option B (

\log \left(10^x+x^{10}\right)+c

) is correct because the derivative of the denominator is exactly the numerator.

649 Easy

649 of 995

Application +4 | -1

\int \frac{x^9}{\left(4 x^2+1\right)^6} d x

is equal to

View answer and explanation

Correct option: D

\frac{1}{10}\left(\frac{1}{\mathrm{x}^2}+4\right)^{-5}+\mathrm{C}

Option D (

\frac{{1}}{{10}}(\frac{{1}}{{x^2}}+4)^{-5}+C

) is correct because simplifying the integral and using substitution leads to this result.

650 Easy

650 of 995

Analyzing +4 | -1

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

is equal to

View answer and explanation

Correct option: D

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

Option D (

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

) is correct because by using integration by parts, the integral simplifies to this expression.

651 Medium

651 of 995

Application +4 | -1

\int \,\,\frac{{dx}}{{\sin \,\,x - \cos \,\,x + \sqrt 2 }}

equals

View answer and explanation

Correct option: C

\frac{1}{{\sqrt 2 }}\,\,\cot \,\,\left( {\frac{x}{2}\,\, + \,\,\frac{\pi }{8}} \right)\,\, + \,\,c

Option D (

- \frac{1}{{\sqrt 2 }}\cot \left( {\frac{x}{2} + \frac{\pi }{8}} \right)\\ + \\,c

) is the correct result after simplifying the integral using trigonometric identities and substitution.

652 Medium

652 of 995

Application +4 | -1

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

, then

\int x^3 f\left(x^2\right) d x

is equal to

View answer and explanation

Correct option: B

\frac{1}{2}\left[x^2 F\left(x^2\right)-\int F\left(x^2\right) d\left(x^2\right)\right]

Option B (

\frac{1}{2}\left[x^2 F\left(x^2\right)-\int F\left(x^2\right) d\left(x^2\right)\right]

) is correct because it uses integration by parts with appropriate substitutions.

653 Easy

653 of 995

Application +4 | -1

I=\int \frac{x^2-1}{x^3 \sqrt{2 x^4-2 x^2+1}} d x

equal to

View answer and explanation

Correct option: D

\frac{\sqrt{2 x^4-2 x^2+1}}{2 x^2}+C

Option D is correct because the integral can be solved by substitution after manipulating the expression.

654 Easy

654 of 995

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 \psi\left(x^3\right)-\int x^2 \psi\left(x^3\right) d x+C

Option C (

\frac{{1}}{{3}} x^3 \psi(x^3)-\int x^2 \psi(x^3) dx+C

) is correct because it represents the result of the given integral after applying substitution and integration by parts.

655 Medium

655 of 995

Application +4 | -1

The value of

\int \frac{\left(\tan ^{-1}(\sin x+1)\right) \cos x}{\left(3+2 \sin x-\cos ^2 x\right)} d x

is (where

\mathrm{c}

is the constant integration)

View answer and explanation

Correct option: C

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

Option C (

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

) is correct because it is the result of the integration after a suitable substitution.

656 Easy

656 of 995

Application +4 | -1

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

View answer and explanation

Correct option: C

\frac{x}{1+\log x}+c

Option C (

\frac{{x}}{{1+\log x}}+c

) is correct because the integral evaluates to this expression after substitution and simplification.

657 Medium

657 of 995

Application +4 | -1

B=\int \frac{1}{e^x+1} d x=-f(x)+C

, where,

C

is the constant of integration and

e^{f(0)}=2

, then the value of

e^{f(-1)}

View answer and explanation

Correct option: B

{e}+1

Option B (e+1) is correct because it is the value obtained after evaluating the integral, finding f(x), and substituting x = -1 into the exponential of f(x).

658 Easy

658 of 995

Application +4 | -1

The value of

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

View answer and explanation

Correct option: C

11\,\,\;\log \,\,\frac{{x - 3}}{{x - 2}}\,\, + \,\,\frac{8}{{x - 2}} + c

Option C (

11\,\;\log \,\,\frac{{x - 3}}{{x - 2}}\,\, + \,\,\frac{8}{{x - 2}} + c

) is correct because it is the result of integrating the given rational function using partial fraction decomposition.

659 Easy

659 of 995

Application +4 | -1

The function

f

whose graph passes through

(4,-20)

and whose derivative is

\cos (\sqrt{4-x})

is given by

View answer and explanation

Correct option: D

-(2 x+12) \sqrt{4-x} \sin \sqrt{4-x} \\ +(4-6 x) \cos \sqrt{4-x}

Option C (

-2(\sqrt{4-x} \sin \sqrt{4-x}+\cos \sqrt{4-x})-18

) is correct because integrating the derivative and using the given point yields this function.

660 Easy

660 of 995

Application +4 | -1

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

is equal to

View answer and explanation

Correct option: A

\frac{{ - \,2}}{{\sqrt {\tan \,\,x} }} + C

Option A (

\frac{{ - \,2}}{{\sqrt {\tan \,\,x} }} + C

) is correct because after rewriting the integrand in terms of tan(x), a simple substitution and power rule integration yields the result.

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

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