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

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

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

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

If $\smallint \,\,{\cos ^4}\,x\,\,dx = Ax\,\, + \,\,B\,\,\sin \,\,2\,x\,\, + \,\,C\,\,\sin \,\,4x\,\, + \,\,D$ , then {A,B,C} equals

$\left\{ {\frac{3}{8},\,\,\frac{1}{{32}}\,\,,\,\,\frac{1}{4}} \right\}$

$\left\{ {\frac{1}{{32}},\,\,\frac{1}{4},\,\,\frac{3}{8}} \right\}$

$\left\{ {\frac{1}{4},\,\,\frac{3}{8},\,\,\frac{1}{{32}}} \right\}$

$\smallint \,{\left( {\frac{{x + 2}}{{x + 4}}} \right)^2}{e^x}\,dx$ is equal to

${e^x}\,\,\left( {\frac{{x\,\, + \,\,2}}{{x\,\, + \,\,4}}} \right)\,\, + \,\,c$

${e^x}\,\,\left( {\frac{{x\,\, - \,\,2}}{{x\,\, + \,\,4}}} \right)\,\, + \,\,c$

$\left( {\frac{{2\,x{e^x}}}{{x\,\, + \,\,4}}} \right)\,\, + \,\,c$

$\smallint \frac{{dx}}{{\left( {x + 1} \right)\,\,\sqrt {4x + 3} }}$ is equal to

$3\,{\tan ^{ - 1}}\,\,\sqrt {4x + 3} \,\, + \,\,c$

$4\,{\tan ^{ - 1}}\,\,\sqrt {4x + 3} \,\, + \,\,c$

$\smallint \,\,\frac{{\cos \,\,x\,\, + \,\,x\,\,\sin \,\,x}}{{{x^2}\,\, + \,\,x\,\,\cos \,\,x}}\,\,\;dx$ is equal to

$\log \,\,\left| {\,\frac{{\sin \,\,x}}{{1\,\, + \,\,\cos \,\,x}}\,} \right| + c$

$\log \,\,\left| {\,\frac{{\sin \,\,x}}{{x\,\, + \,\,\cos \,\,x}}\,} \right| + c$

$\log \,\,\left| {\,\frac{{2\,\,\sin \,\,x}}{{x\,\, + \,\,\cos \,\,x}}\,} \right| + c$

The value of $\int \frac{1+\sin x}{1-\sin x} d x$ is

$2 \tan \left(\frac{x}{2}+\frac{\pi}{4}\right)+C$

$2 \tan ^2\left(\frac{x}{2}+\frac{\pi}{4}\right)-x+C$

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

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

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

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

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JEE • MATHEMATICS • INDEFINITE INTEGRATION

Indefinite integration practice · Mixed difficulty

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Showing 751-780 of 995 questions

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751 Easy

751 of 995

Application +4 | -1

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

is equal to

View answer and explanation

Correct option: B

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

Option B (

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

) is correct because it is the result of simplifying the integrand and using appropriate integration techniques.

752 Medium

752 of 995

Application +4 | -1

\int \,\,{\cos ^4}\,x\,\,dx = Ax\,\, + \,\,B\,\,\sin \,\,2\,x\,\, + \,\,C\,\,\sin \,\,4x\,\, + \,\,D

, then {A,B,C} equals

View answer and explanation

Correct option: B

\left\{ {\frac{3}{8},\,\,\frac{1}{4},\,\,\frac{1}{{32}}} \right\}

Option B (

\left\{{{\frac{3}{8},\\,\frac{1}{4},\\,\frac{1}{{32}}}}\right\}

) is correct because the integral of

\cos^4 x

is evaluated using trigonometric identities and then the coefficients are matched.

753 Medium

753 of 995

Analyzing +4 | -1

\int \,{\left( {\frac{{x + 2}}{{x + 4}}} \right)^2}{e^x}\,dx

is equal to

View answer and explanation

Correct option: A

{e^x}\,\,\left( {\frac{x}{{x\,\, + \,\,4}}} \right)\,\, + \,\,c

Option A is correct because the integral simplifies to

{e^x}\,\left( {\frac{x}{{x\,\, + \,\,4}}} \right)\,\, + \,\,c

after applying integration by parts and simplification.

754 Easy

754 of 995

Application +4 | -1

\int \frac{{dx}}{{\left( {x + 1} \right)\,\,\sqrt {4x + 3} }}

is equal to

View answer and explanation

Correct option: C

2\,{\tan ^{ - 1}}\,\,\sqrt {4x + 3} \,\, + \,\,c

Option C (

2\,{\tan ^{ - 1}}\,\sqrt {4x\, + \,3} \, + \,c

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

755 Easy

755 of 995

Understanding +4 | -1

\int \,\,\frac{{\cos \,\,x\,\, + \,\,x\,\,\sin \,\,x}}{{{x^2}\,\, + \,\,x\,\,\cos \,\,x}}\,\,\;dx

is equal to

View answer and explanation

Correct option: D

\log \,\,\left| {\,\frac{x}{{x\,\, + \,\,\cos \,\,x}}\,} \right| + c

Option D,

\log \,\left| {\,\frac{x}{{x\,+\cos \,x}}\} \right| + c

, is correct because the integral simplifies to this form after algebraic manipulation and recognizing the derivative in the denominator.

756 Easy

756 of 995

Application +4 | -1

The value of

\int \frac{1+\sin x}{1-\sin x} d x

View answer and explanation

Correct option: C

2 \tan \left(\frac{x}{2}+\frac{\pi}{4}\right)-x+C

Option C (

2 \tan \left(\frac{x}{2}+\frac{\pi}{4}\right)-x+C

) is correct because it is the result of integrating the given expression and simplifying using trigonometric identities.

757 Easy

757 of 995

Analyzing +4 | -1

\int \frac{\cos x-\sin x}{\cos x+\sin x}(2+2 \sin 2 x) d x

is equal to

View answer and explanation

Correct option: A

\sin 2 x+C

Option A (

\sin 2 x+C

) is correct because the integral simplifies to

2 \int \cos 2x dx

, which equals

\sin 2x + C

758 Easy

758 of 995

Application +4 | -1

\int e^x \tan ^2\left(e^x\right) d x=

View answer and explanation

Correct option: D

\tan \left(e^x\right)-e^x+c

Option D (

\tan \left(e^x\right)-e^x+c

) is correct because the integral simplifies to this form after substitution and integration.

759 Medium

759 of 995

Analyzing +4 | -1

Let

f

be a polynomial function such that for all real

x, f\left(x^2+1\right)=x^4+5 x^2+2

, then

\int f(x) d x

View answer and explanation

Correct option: A

\frac{x^3}{3}+\frac{3 x^2}{2}-2 x+c

Option A,

\frac{x^3}{3}+\frac{3 x^2}{2}-2 x+c

, is correct because by substituting

x^2+1

into f, we can determine f(x) and then integrate it.

760 Medium

760 of 995

Analyzing +4 | -1

\int \frac{\sin x-\cos x}{(\sin x+\cos x) \sqrt{\sin x \cos x+\sin ^2 x \cos ^2 x}} d x=

View answer and explanation

Correct option: C

-\operatorname{Sec}^{-1}(\sin 2 x+1)+c

The correct answer is C (

-\operatorname{Sec}^{-1}(\sin 2 x+1)+c

) because the integral simplifies to a standard inverse secant form after appropriate trigonometric manipulations and substitution.

761 Medium

761 of 995

Application +4 | -1

\int \frac{a \sin x+b \cos x}{c \sin x+d \cos x} d x=p x+q \log _e|c \sin x+d \cos x|+c

where

p+q

is equal to

View answer and explanation

Correct option: A

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

Option A (

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

) is correct because it represents the value of

p+q

obtained by decomposing the integrand and integrating.

762 Easy

762 of 995

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.

763 Easy

763 of 995

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.

764 Easy

764 of 995

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.

765 Easy

765 of 995

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

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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

766 Easy

766 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,

\tan (1+\log x)+c

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

767 Medium

767 of 995

Application +4 | -1

F_n(x)=\int(\log x)^n d x

and

\phi_n=F_n+n F_{n-1}

, then

\phi_n(e)-\phi_n(1)

is equal to

View answer and explanation

Correct option: D

e

Option D is correct because by using integration by parts, we find that φ_n(x) = x(log x)^n + c, and then φ_n(e) - φ_n(1) = e.

768 Easy

768 of 995

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

769 Medium

769 of 995

Application +4 | -1

Let

f

be a real-valued function defined on the interval

\left( { - 1,\,1} \right)

such that

{e^{ - x}}\,\,f\,\,\left( x \right) = 2\,\, + \,\,\int \nolimits_0^x \,\,\sqrt {{t^4}\,\, + \,\,1} \,\,dt

, for all

x \in \left( { - 1,\,\,1} \right)

and let

{f^{ - 1}}

be the inverse function of

f

. Then,

{\left( {{f^{ - 1}}} \right)^\prime }\,\,\left( 2 \right)

is equal to

View answer and explanation

Correct option: B

\frac{1}{3}

Option B (1/3) is correct because after differentiating and using the inverse function property, the derivative of the inverse function at 2 is found to be 1/3.

770 Easy

770 of 995

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.

771 Easy

771 of 995

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.

772 Easy

772 of 995

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.

773 Medium

773 of 995

Application +4 | -1

Let

f(x)=\int \frac{\sqrt{x}}{(1+x)^2} d x(x \geq 0)

. Then

f(3)-f(1)

is equal to :

View answer and explanation

Correct option: D

\frac{\pi}{12}+\frac{1}{2}-\frac{\sqrt{3}}{4}

Option D

\frac{\pi}{12}+\frac{1}{2}-\frac{\sqrt{3}}{4}

is correct because it is the result of evaluating the definite integral after appropriate substitution and simplification.

774 Easy

774 of 995

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.

775 Easy

775 of 995

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.

776 Easy

776 of 995

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.

777 Easy

777 of 995

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.

778 Medium

778 of 995

Application +4 | -1

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

equals

View answer and explanation

Correct option: A

\frac{1}{2} \log _e|\sin x-\cos x|+\frac{x}{2}+c

Option A is correct because the integral is solved by rewriting the numerator and separating the integral into simpler parts, resulting in (1/2)log_e|sinx-cosx| + x/2 + c.

779 Easy

779 of 995

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.

780 Easy

780 of 995

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)]

Revision notes and key takeaways

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