The value of $\smallint \,\,\frac{{\sec \,\,x\,\,dx}}{{\sqrt {\sin \,\,\left( {2x + {\rm{\theta }}} \right)\,\, + \,\,\sin \,\,{\rm{\theta }}} }}$ is

$\sqrt {2\,\,(\tan \,\,x\,\, + \,\,\tan \,\,{\rm{\theta }})\,\,\sec \,\,{\rm{\theta }}} + c$

$\sqrt {(\tan \,\,x\,\, + \,\,\tan \,\,{\rm{\theta }})\,\,\sec \,\,{\rm{\theta }}} + c$

$\sqrt {2\,\,(\sin \,\,\,x\,\, + \,\,\tan \,\,{\rm{\theta }})\,\,\sec \,\,{\rm{\theta }}} + c$

Correct evaluation of $\int \frac{x}{(x-2)(x-1)} d x$ is

$\log _e \frac{(x-1)}{(x-2)}+p$

$2 \log _e\left(\frac{x-2}{x-1}\right)+p$

The value of the integral $\smallint \,\,\frac{{x\,\,\sin \,\,{x^2}\,{e^{\sec {x^2}}}}}{{{{\cos }^2}\,\,{x^2}}}\,\,dx$, is

$\frac{1}{2}\,\,{e^{\sin {x^2}}} + \,\,C$

$\frac{1}{2}\,\,\sin \,\,{x^2}\,{e^{{{\cos }^2}{x^2}}} + \,\,C$

$\smallint \,\,\frac{{dx}}{{1 - \cos \,\,x - \sin \,\,x}}$ is equal to

$\log \,\,\left| {1 - \tan \,\,\frac{x}{2}} \right| + c$

$\log \,\,\left| {1 + \tan \,\,\frac{x}{2}} \right| + c$

$\smallint \,\,\cos \,\,\left\{ {2\,\,{{\tan }^{ - 1}}\,\,\sqrt {\frac{{1 - x}}{{1 + x}}\,} } \right\}\,\,dx$ is equal to

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

$\smallint \,\,\frac{{{x^2} + \,\,x\,\, - 6}}{{\left( {x - 2} \right)\,\,\left( {x - 1} \right)}}\,\,dx$

$x\,\, + \,\,2\,\,\log \,\,\left( {x - 1} \right)\,\, + \,\,c$

$2x\,\, + \,\,2\,\,\log \,\,\left( {x - 1} \right)\,\, + \,\,c$

$x\,\, + \,\,4\,\,\log \,\,\left( {1 - x} \right)\,\, + \,\,c$

$\smallint \,\,\frac{{dx}}{{(\sin \,\,x\,\, - \,\,2\,\,\cos \,\,x)\,(2\,\,\sin \,\,x\,\, + \,\,\cos \,\,x)}}$ is equal to

$\frac{1}{4}\,\,\log \,\,\left( {\frac{{\tan \,\,x\,\, - \,\,2}}{{\tan \,\,x\,\, + \,\,2}}} \right)\,\, + \,\,c$

$ - \,\,\frac{1}{4}\,\,\log \,\,\left( {\frac{{2\,\,\sin \,\,x\,\, + \,\,\cos \,\,x}}{{\sin \,\,x\,\, + \,\,2\,\,\cos \,\,x}}} \right) + c$

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JEE • MATHEMATICS • INDEFINITE INTEGRATION

Indefinite integration practice · Mixed difficulty

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Showing 841-870 of 995 questions

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841 Medium

841 of 995

Analyzing +4 | -1

The value of

\int \,\,\frac{{\sec \,\,x\,\,dx}}{{\sqrt {\sin \,\,\left( {2x + \mathrm{\theta }} \right)\,\, + \,\,\sin \,\,\mathrm{\theta }} }}

View answer and explanation

Correct option: B

\sqrt {2\,\,(\tan \,\,x\,\, + \,\,\tan \,\,\mathrm{\theta })\,\,\sec \,\,\mathrm{\theta }} + c

Option B (

\sqrt {2\,\,(\tan \,\,x\,\, + \,\,\tan \,\,\mathrm{\theta })\,\,\sec \,\,\mathrm{\theta }} + c

) is correct because it is the result of correctly evaluating the integral after trigonometric simplification and substitution.

842 Easy

842 of 995

Analyzing +4 | -1

\int \frac{e^{\tan ^{-1} x}}{\left(1+x^2\right)}\left[\left(\sec ^{-1} \sqrt{1+x^2}\right)^2+\cos ^{-1}\left(\frac{1-x^2}{1+x^2}\right)\right] d x,

(x>0)

is equal to

View answer and explanation

Correct option: C

e^{\tan ^{-1} x} \cdot\left(\sec ^{-1}\left(\sqrt{1+x^2}\right)\right)^2+C

Option C (

e^{\tan^{-1} x} \cdot (\sec^{-1}(\sqrt{{1 + x^2}}))^2 + C

) is correct because the integral simplifies to this form after applying trigonometric identities and substitution.

843 Easy

843 of 995

Application +4 | -1

Correct evaluation of

\int \frac{x}{(x-2)(x-1)} d x

View answer and explanation

Correct option: A

\log _e \frac{(x-2)^2}{(x-1)}+p

Option A,

\log _e \frac{(x-2)^2}{(x-1)}+p

, is the correct evaluation because it results from partial fraction decomposition and integration of the given expression.

844 Medium

844 of 995

Application +4 | -1

The value of integral

\int \nolimits_0^1 \,\sqrt {\frac{{1\,\, - \,\,x}}{{1\,\, + \,\,x}}} \;\,dx

View answer and explanation

Correct option: B

\frac{\pi }{2}\,\, - \,\,1

Option B (

\frac{\pi }{2}\\ - \\1

) is correct because the integral can be solved by rationalizing the denominator and then using trigonometric substitution or direct integration.

845 Easy

845 of 995

Application +4 | -1

The value of

\int \frac{d x}{\sqrt{(x-a)(b-x)}}

, is

View answer and explanation

Correct option: A

2 \sin ^{-1} \sqrt{\frac{x-a}{b-a}}+C

Option A (

2 \sin ^{-1} \sqrt{\frac{x-a}{b-a}}+C

) is correct because it is the result of the integral after applying the trigonometric substitution

x=a \cos ^2 \theta+b \sin ^2 \theta

846 Medium

846 of 995

Application +4 | -1

The value of the integral

\int \,\,\frac{{x\,\,\sin \,\,{x^2}\,{e^{\sec {x^2}}}}}{{{{\cos }^2}\,\,{x^2}}}\,\,dx

, is

View answer and explanation

Correct option: A

\frac{1}{2}\,\,{e^{\sec {x^2}}} + \,\,C

Option A (

\frac{1}{2}\,{e^{\sec {x^2}}} + \,C

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

847 Medium

847 of 995

Application +4 | -1

\int \,\,\frac{{dx}}{{1 - \cos \,\,x - \sin \,\,x}}

is equal to

View answer and explanation

Correct option: C

\log \,\,\left| {1 - \cot \,\,\frac{x}{2}} \right| + c

Option C is correct because the integral simplifies to

\log \,\,\left| {1 - \cot \,\,\frac{x}{2}} \right| + c

after using trigonometric identities and substitution.

848 Easy

848 of 995

Application +4 | -1

\int \,\,\cos \,\,\left\{ {2\,\,{{\tan }^{ - 1}}\,\,\sqrt {\frac{{1 - x}}{{1 + x}}\,} } \right\}\,\,dx

is equal to

View answer and explanation

Correct option: D

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

Option D (

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

) is correct because after simplifying the trigonometric expression, the integral becomes

\int x dx

, which evaluates to

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

849 Medium

849 of 995

Application +4 | -1

If the primitive of

\frac{1}{{f\left( x \right)}}

is equal to

\log \,\,{\left\{ {f\left( x \right)} \right\}^2}\,\, + \,\,c,\,\,\mathrm{then}\,\,f\left( x \right)

View answer and explanation

Correct option: B

\frac{x}{2}\,\, + \,\,d

Option B (

\frac{x}{2}\, + \,d

) is correct because differentiating

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

gives

\frac{1}{{f\left( x \right)}}

when

f(x) = \frac{x}{2} + d

850 Easy

850 of 995

Application +4 | -1

\int \,\,\frac{{{x^2} + \,\,x\,\, - 6}}{{\left( {x - 2} \right)\,\,\left( {x - 1} \right)}}\,\,dx

View answer and explanation

Correct option: D

x\,\, + \,\,4\,\,\log \,\,\left( {x - 1} \right)\,\, + \,\,c

Option D (

x\\,\\ + \\,4\\,\log \\,\left( {x - 1} \right)\\,\\ + \\,\\,c

) is correct because the integral simplifies to a rational function that can be easily integrated after partial fraction decomposition.

851 Easy

851 of 995

Application +4 | -1

\int \frac{(x+3) e^x}{(x+4)^2} d x=

View answer and explanation

Correct option: C

\frac{e^x}{x+4}+c

Option C (

\frac{{e^x}}{{x+4}}+c

) is correct because the integral can be simplified and solved using integration by parts or by recognizing the form

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

852 Medium

852 of 995

Application +4 | -1

The value of

\int \nolimits_0^{\mathrm{\pi }/2} \,\,\frac{{\cos \,\,\mathrm{\theta }}}{{\sqrt {4\,\, - \,\,{{\sin }^2}\,\,\mathrm{\theta }} }}\,\,d\mathrm{\theta }

View answer and explanation

Correct option: B

\frac{\mathrm{\pi }}{6}

Option B (

\frac{\mathrm{\pi }}{6}

) is correct because after substitution, the integral becomes a standard arcsin integral, evaluating to

\frac{\pi}{6}

853 Easy

853 of 995

Application +4 | -1

\int \frac{3 x^2}{\sqrt{9-16 x^6}} d x=

View answer and explanation

Correct option: A

\frac{1}{4} \sin ^{-1}\left(\frac{4 x^3}{3}\right)+c

Option A is correct because the integral can be solved using substitution, resulting in

\frac{1}{4} \sin^{-1}(\frac{4x^3}{3}) + c

854 Easy

854 of 995

Application +4 | -1

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

where

C

is constant of integration. Then

f(x)

is equal to:

View answer and explanation

Correct option: C

x

Option C (

x

) is correct because integrating the given expression using substitution

\cot^{-1}x = t

leads to

f(x) = x

855 Medium

855 of 995

Analyzing +4 | -1

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

equals

View answer and explanation

Correct option: A

\frac{\sqrt{x^4+x^2+1}}{x}+c

Option A is correct because the integral simplifies to a form that can be solved by substitution, resulting in the given expression.

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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

856 Easy

856 of 995

Application +4 | -1

\int \frac{\sin x+8 \cos x}{4 \sin x+6 \cos x} d x=

View answer and explanation

Correct option: A

x+\frac{1}{2} \log |4 \sin x+6 \cos x|+C

Option C (

x+2 \log |2 \sin x+3 \cos x|+C

) is correct because the integral is solved by expressing the numerator as a linear combination of the denominator and its derivative.

857 Easy

857 of 995

Application +4 | -1

I=\int \cos \left\{2 \tan ^{-1} \sqrt{\frac{1-x}{1+x}}\right\} d x

is equal to

View answer and explanation

Correct option: A

\frac{1}{2} x^2+C

Option A is correct because the integral can be simplified using trigonometric identities and a substitution.

858 Easy

858 of 995

Application +4 | -1

f: \mathbb{R} \rightarrow \mathbb{R}

is a function such that

f(x)+2 f\left(\frac{1}{x}\right)=3 x \,\forall x \neq 0

then

\int x f(x) dx

is equal to

View answer and explanation

Correct option: A

2x - \frac{1}{3}{x^3} + c

Option A (

2x - \frac{1}{3}{x^3} + c

) is correct because solving the functional equation for

f(x)

and then integrating

xf(x)

yields this result.

859 Easy

859 of 995

Application +4 | -1

\int e^{-x} \operatorname{cosec}^2\left(2 e^{-x}+5\right) d x=

View answer and explanation

Correct option: A

\frac{1}{2} \cot \left(2 e^{-x}+5\right)+c

Option A (

\frac{1}{2} \cot \left(2 e^{-x}+5\right)+c

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

860 Easy

860 of 995

Application +4 | -1

\int \frac{x-1}{(x-3)(x-2)} d x=

View answer and explanation

Correct option: B

\log (x-3)^2-\log (x-2)+c

Option B (

\log (x-3)^2-\log (x-2)+c

) is correct because the integral is solved by partial fraction decomposition.

861 Medium

861 of 995

Application +4 | -1

f\left(\frac{x-4}{x+2}\right)=2 x+1,(x \in R=\{1,-2\})

, then

\int f(x) d x

is equal to (where

C

is a constant of integration)

View answer and explanation

Correct option: B

-12 \log _e|1-x|-3 x+c

Option B (

-12 \log _e|1-x|-3 x+c

) is correct because it is the result of integrating f(x) after finding the expression for f(x) by substituting

y = \frac{x-4}{x+2}

862 Easy

862 of 995

Application +4 | -1

Let

n \ge 2

be a natural number and

0 < \theta < \frac{\pi }{2}.

Then

\int \frac{{{{({{\sin }^n}\theta - \sin \theta )}^{\frac{1}{n}}}\cos \theta }}{{{{\sin }^{n + 1}}\theta }}d\theta

is equal to (where C is a constant of integration)

View answer and explanation

Correct option: C

\frac{n}{{{n^2} - 1}}{\left( {1 - \frac{1}{{{{\sin }^{n - 1}}\theta }}} \right)^{\frac{{n + 1}}{n}}} + C

Option D (1356) is correct because it is the sum of all two-digit numbers that leave a remainder of 2 or 5 when divided by 7.

863 Easy

863 of 995

Application +4 | -1

The value of

\int \frac{1}{x^n\left(1+x^n\right)^{1 / n}} d x,(n \in N)

, is

View answer and explanation

Correct option: A

\frac{1}{1-n}\left\{1+\frac{1}{x^n}\right\}^{1-\frac{1}{n}}+C

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

864 Easy

864 of 995

Application +4 | -1

\int e^x \cos 4 x \mathrm{~d} x=A e^{5 x}\left(\sin 4 x+\frac{5}{4} \cos 4 x\right)+C

then

A

is equal to

View answer and explanation

Correct option: A

\frac{4}{41}

Option A is correct because after integrating by parts twice and solving for the integral, the coefficient A is found to be

\frac{4}{41}

865 Medium

865 of 995

Application +4 | -1

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

is equal to

View answer and explanation

Correct option: A

{\log _e}\,\,\sqrt[5]{{\frac{{\tan \,\,x\,\, - \,\,2}}{{2\tan \,\,x\,\, + \,\,1}}}}\,\, + \,\,c

Option A (

{\log _e}\sqrt[5]{{\frac{{\tan \\,x - 2}}{{2\tan \\,x + 1}}}}\\ + \\ c

) is correct because the integral is solved by dividing by cos²(x) and using partial fractions.

866 Easy

866 of 995

Application +4 | -1

\int \log (x+1) d x=

View answer and explanation

Correct option: A

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

Option A (

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

) is correct because it is the result of integrating

\log(x+1)

using integration by parts.

867 Easy

867 of 995

Analyzing +4 | -1

\int \frac{d x}{\cos (2 x) \cos (4 x)}

is equal to

View answer and explanation

Correct option: A

\frac{1}{2 \sqrt{2}} \log \left|\frac{1+\sqrt{2} \sin 2 x}{1-\sqrt{2} \sin 2 x}\right|-\frac{1}{2}(\log |\sec 2 x-\tan 2 x|)+C

Option A is correct because after using trigonometric identities and integrating, the result matches the given expression.

868 Easy

868 of 995

Analyzing +4 | -1

The value of

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

, is

View answer and explanation

Correct option: A

-\frac{1}{2 \sqrt{3}} \log \left|\frac{\sqrt{x^4+x^2+1}-\sqrt{3} x}{\sqrt{x^4+x^2+1}+\sqrt{3} x}\right|+C

Option A (

-\frac{1}{2 \sqrt{3}} \log \left|\frac{\sqrt{x^4+x^2+1}-\sqrt{3} x}{\sqrt{x^4+x^2+1}+\sqrt{3} x}\right|+C

) is correct because the integral simplifies to this logarithmic form after appropriate substitutions.

869 Easy

869 of 995

Analyzing +4 | -1

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

, then

\int\{f(x)\}^2 d x

is equal to

View answer and explanation

Correct option: A

\frac{1}{2}\{f(x)\}^2

Option A (

\frac{1}{2}{f(x)}^2

) is correct because if

\int f(x) dx = f(x)

, then

f(x) = Ce^x

, and

\int {f(x)}^2 dx = \frac{1}{2}{f(x)}^2

870 Easy

870 of 995

Application +4 | -1

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

, then

\int\{f(x)\}^2 d x

is equal to

View answer and explanation

Correct option: A

\frac{1}{2}\{f(x)\}^2

Option A (

\frac{1}{2}\{f(x)\}^2

) is correct because the integral of the square of the function is half the square of the function, given the initial condition.

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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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