The value of $\mathop \smallint \limits_\alpha ^\beta \,x\,\;\left| {\,x\,} \right|\,\,dx$, where $\alpha < 0 < \beta $, is

$\frac{1}{2}\,\,\left( {{\alpha ^2} + {\beta ^2}} \right)$

$\frac{1}{3}\,\,\left( {{\beta ^2} - {\alpha ^2}} \right)$

The value of the integral $\mathop \smallint \limits_{ - 1}^1 \,\,{\sin ^{11}}\,\,x\,\,dx$, is

$\frac{{10}}{{11}},\,\,\frac{8}{9},\,\,\frac{6}{7},\,\,\frac{4}{5},\,\,\frac{2}{3}$

$\frac{{10}}{{11}},\,\,\frac{8}{9},\,\,\frac{6}{7},\,\,\frac{4}{5},\,\,\frac{2}{3},\,\,\frac{\pi }{2}$

If $f\left( x \right) = \mathop \smallint \nolimits_0^x \,\,({\sin ^4}t\,\, + \,\,{\cos ^4}\,\,t)\,dt$, then \[f\,\,\left( {x\,\, + \,\,{\rm{\pi }}} \right)\] will be equal to

$f\left( x \right)\,\, + \,\,f\left( {\rm{\pi }} \right)$ or $f\left( x \right)\,\, + \,\,2f\left( {\frac{{\rm{\pi }}}{2}} \right)$

$f\left( x \right)\,\, + \,\,f\left( {\frac{{\rm{\pi }}}{2}} \right)$

$f\left( x \right)\,\, - \,\,f\left( {\rm{\pi }} \right)$

$f\left( x \right)\,\, - \,\,2\,f\left( {\frac{{\rm{\pi }}}{2}} \right)$

The value of $\mathop \smallint \limits_{ - \pi }^\pi \left( {1 - {x^2}} \right)\,\,\sin \,\,x\,\,{\cos ^2}\,\,x\,\,dx$, is

$\pi \,\, - \,\,\frac{{{\pi ^3}}}{3}$

$\frac{\pi }{2}\,\, - \,\,2{\pi ^3}$

If $f\,\,\left( x \right) = f\,\,\left( {a - x} \right)\,\,{\rm{and}}\,\,{\rm{g}}\left( x \right)\,\, + \,\,{\rm{g}}\left( {a - x} \right) = 2$,then the value of $\mathop \smallint \nolimits_0^a \,\,f\left( x \right)\,\,{\rm{g}}\,\,\left( x \right)\,\,dx\,\,{\rm{is}}$ is

$\mathop \smallint \nolimits_0^a \,\,f\,\,\left( x \right)\,\,dx$

$\mathop \smallint \nolimits_0^a \,\,g\,\,\left( x \right)\,\,dx$

$\mathop \smallint \nolimits_0^a \,\,[\,{\rm{g}}\left( x \right)\,\, - \,\,f\left( x \right)\,\,dx$

$\mathop \smallint \nolimits_0^a \,\,[\,{\rm{g}}\left( x \right)\,\, + \,\,f\left( x \right)\,\,dx$

$\mathop \smallint \nolimits_0^{\pi /4} \,\,\log \,\,(1 + \tan \,\,x)\,\,dx$ is equal to

$\frac{\pi }{4}\,\,{\log _2}\,\,e$

$\frac{\pi }{4}\,\,{\log _e}\,\,2$

$\frac{\pi }{8}\,\,{\log _e}\,\,\left( {\frac{1}{2}} \right)$

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JEE • MATHEMATICS • DEFINITE INTEGRATION

Definite integration practice · Mixed difficulty

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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.

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751 Easy

751 of 995

Application +4 | -1

The area bounded by the straight lines

x=0, x=2

and the curves

y=2^x, y=2 x-x^2

View answer and explanation

Correct option: D

\frac{3}{\log 2}-\frac{4}{3}

Option D (

\frac{3}{\log 2}-\frac{4}{3}

) is correct because it represents the definite integral of the difference between the two functions between the specified limits.

752 Easy

752 of 995

Application +4 | -1

The value of

\int_0^\pi \frac{e^{\cos x} \sin x}{\left(1+\cos ^2 x\right)\left(e^{\cos x}+e^{-\cos x}\right)} d x

is equal to

View answer and explanation

Correct option: C

\frac{\pi}{4}

Option C (

\frac{\pi}{4}

) is correct because it is the result of evaluating the definite integral using the property of definite integrals and substitution.

753 Easy

753 of 995

Application +4 | -1

\int \nolimits_0^{\pi /2} \,\,\frac{{\cos \,\,x - \,\,\sin \,\,x}}{{1\,\, + \,\cos \,\,x\,\,\sin \,\,x}}\,\,dx

is equal to

View answer and explanation

Correct option: A

\mathrm{0}

Option A (0) is correct because using the property of definite integrals, the integral evaluates to zero.

754 Medium

754 of 995

Application +4 | -1

The value of

\int \nolimits_1^2 \,\,\frac{{dx}}{{\left( {x + 1} \right)\,\,\sqrt {{x^2} - 1} }}

View answer and explanation

Correct option: B

\frac{1}{{\sqrt 3 }}

The correct answer is 1/√(3) because after substituting x = sec θ and evaluating the definite integral, we obtain this value.

755 Easy

755 of 995

Understanding +4 | -1

S_n=\frac{e^{1 / n}}{n^2}+\frac{2}{n^2} e^{2 / n}+\frac{3}{n^2} e^{3 / n}+\cdots+\frac{1}{n} e

then

\lim \limits_{{n \rightarrow \infty}} S_n

View answer and explanation

Correct option: A

Option A (1) is correct because the limit of the given sum can be expressed as a definite integral which evaluates to 1.

756 Easy

756 of 995

Application +4 | -1

The value of

\int \limits_\alpha ^\beta \,x\,\;\left| {\,x\,} \right|\,\,dx

, where

\alpha < 0 < \beta

, is

View answer and explanation

Correct option: C

\frac{1}{3}\,\,\left( {{\alpha ^3} + {\beta ^3}} \right)

Option D (None of these) is incorrect; the correct answer is

\frac{1}{3}\,\,\left( {{\alpha ^3} + {\beta ^3}} \right)

, derived by splitting the integral and considering the absolute value.

757 Easy

757 of 995

Application +4 | -1

y=f(x)

be an invertible function with inverse

g

and

h(x)=x f(x)

, then

\int\limits_{{f(a)}}^{f(b)} g(x) d x+\int\limits_{a}^b f(x) d x

is equal to

View answer and explanation

Correct option: B

h(b)-h(a)

Option B (h(b)-h(a)) is correct because after substitution and integration by parts, the expression simplifies to this.

758 Easy

758 of 995

Analyzing +4 | -1

\int\limits_{0}^{\pi / 2} \sin 2 x \log \tan x \,d x

is equal to

View answer and explanation

Correct option: C

Option C (0) is correct because the integral evaluates to zero due to the properties of the logarithm and trigonometric functions after applying a substitution.

759 Easy

759 of 995

Application +4 | -1

Let

f\left( x \right) = x - \left[ x \right]

, for every real

x

, where [

x

]is the greatest integer less than or equal to

x

. Then,

\int \nolimits_{ - 1}^1 \,\,f\,\,\left( x \right)dx\,\,\mathrm{is}

View answer and explanation

Correct option: A

\mathrm{1}

Option A (

\mathrm{1}

) is correct because the integral of the fractional part of x over the interval [-1, 1] equals 1.

760 Easy

760 of 995

Application +4 | -1

Let

\frac{d}{{dx}}\;\,\,F\left( x \right) = \left( {\frac{{{e^{\sin x}}}}{x}} \right),\,\,x > 0

, If

\int \nolimits_1^4 \,\,\frac{3}{x}\,\,{e^{\sin {x^3}}}\;dx = F\left( k \right)\,\, - F\left( 1 \right)

, then one of the possible values of

k

, is

View answer and explanation

Correct option: D

\mathrm{64}

The correct answer is D (64) because after substitution and simplification, the value of k is found to be 64.

761 Easy

761 of 995

Application +4 | -1

Let function

F

be defined as

F(x)=\int\limits_{1}^x \frac{\mathrm{e}^t}{\mathrm{t}} \mathrm{dt}, x > 0

then the value of the integral

\int\limits_{1}^x \frac{\mathrm{e}^t}{t+a} d t

, where

a > 0

, is:

View answer and explanation

Correct option: D

e^{-a}[F(x+a)-F(1+a)]

Option D (

e^{-a}[F(x+a)-F(1+a)]

) is correct because it is the result obtained after applying a suitable substitution and using the given definition of the function F(x).

762 Easy

762 of 995

Understanding +4 | -1

The value of the integral

\int \limits_{ - 1}^1 \,\,{\sin ^{11}}\,\,x\,\,dx

, is

View answer and explanation

Correct option: D

\mathrm{0}

Option D (0) is correct because

\sin^{11}(x)

is an odd function, and the integral of an odd function over a symmetric interval is zero.

763 Hard

763 of 995

Application +4 | -1

f\left( x \right) = \int \nolimits_0^x \,\,({\sin ^4}t\,\, + \,\,{\cos ^4}\,\,t)\,dt

, then

f\,\,\left( {x\,\, + \,\,\mathrm{\pi }} \right)

will be equal to

View answer and explanation

Correct option: B

f\left( x \right)\,\, + \,\,f\left( \mathrm{\pi } \right)

f\left( x \right)\,\, + \,\,2f\left( {\frac{\mathrm{\pi }}{2}} \right)

Option B (

f(x) + f(\pi)

f(x) + 2f(\frac{{\pi }}{2})

) is correct because the function is periodic and the integral can be split into parts.

764 Easy

764 of 995

Understanding +4 | -1

\int \nolimits_{ - 1}^1 \,\,{\sin ^3}\,\,x\,\,{\cos ^2}\,\,x\,dx

is equal to

View answer and explanation

Correct option: C

\mathrm{0}

Option C (0) is correct because the integrand is an odd function, and the integral is evaluated over a symmetric interval [-1, 1].

765 Easy

765 of 995

Application +4 | -1

\int \,\,{e^{ - \log x}}\,\,\;dx

is equal to

View answer and explanation

Correct option: D

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

Option D (

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

) is correct because the integral of

e^{-\log x}

simplifies to the integral of 1/x, which is log|x| + c.

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SEQUENCE OF SERIES 1764 MCQs MATRICES 1363 MCQs INDEFINITE INTEGRATION 1179 MCQs PROBABILITY 1146 MCQs COMPLEX NUMBERS 980 MCQs DIFFERENTIAL EQUATIONS 977 MCQs

766 Easy

766 of 995

Application +4 | -1

The value of the integral

\int \nolimits_3^6 \,\,\frac{{\sqrt x }}{{\sqrt {9 - x} + \,\,\sqrt x }}\,\,dx

View answer and explanation

Correct option: A

\frac{3}{2}

Option A is correct because the integral evaluates to

\frac{3}{2}

after using the property of definite integrals.

767 Medium

767 of 995

Application +4 | -1

The integral

\int \frac{\mathrm{dx}}{(x+1)^{\frac{3}{4}}(x-2)^{\frac{5}{4}}}

is equal to:

View answer and explanation

Correct option: B

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

Option B (

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

) is correct because the integral can be solved by substitution, letting

t = \frac{{x+1}}{{x-2}}

768 Medium

768 of 995

Understanding +4 | -1

The value of

\int \limits_{ - \pi }^\pi \left( {1 - {x^2}} \right)\,\,\sin \,\,x\,\,{\cos ^2}\,\,x\,\,dx

, is

View answer and explanation

Correct option: A

Option A is correct because the integrand is an odd function, and the integral of an odd function over a symmetric interval is zero.

769 Easy

769 of 995

Application +4 | -1

f\,\,\left( x \right) = f\,\,\left( {a - x} \right)\,\,\mathrm{and}\,\,\mathrm{g}\left( x \right)\,\, + \,\,\mathrm{g}\left( {a - x} \right) = 2

,then the value of

\int \nolimits_0^a \,\,f\left( x \right)\,\,\mathrm{g}\,\,\left( x \right)\,\,dx\,\,\mathrm{is}

View answer and explanation

Correct option: A

\int \nolimits_0^a \,\,f\,\,\left( x \right)\,\,dx

Option A (

\int \nolimits_0^a \,\,f\,\,\left( x \right)\,\,dx

) is correct because by substituting x with a-x and using the given properties, the integral simplifies to this form.

770 Medium

770 of 995

Analyzing +4 | -1

The integral

\int\limits_{0}^{1 / a} \frac{\log (1+a x)}{1+a^2 x^2} \mathrm{~d} x(a>0)

is equal to

View answer and explanation

Correct option: C

\frac{1}{a}(\log 2) \frac{\pi}{8}

Option C (

\frac{1}{a}(\log 2) \frac{\pi}{8}

) is correct because after substitution and integration, the integral evaluates to

\frac{1}{a}(\log 2) \frac{\pi}{8}

771 Easy

771 of 995

Application +4 | -1

f(\alpha)=\int_1^\alpha \frac{\log _{10} t}{1+t} d t, \alpha>0

, then

f\left(e^3\right)+f\left(e^{-3}\right)

is equal to:

View answer and explanation

Correct option: D

\frac{9}{2 \log _e(10)}

Option D (

\frac{{9}}{{2 \log_e(10)}}

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

772 Easy

772 of 995

Application +4 | -1

The value of

\int\limits_{0}^1 \frac{8 \log (1+x)}{1+x^2} d x

View answer and explanation

Correct option: D

\pi \log 2

Option D (

\pi \log 2

) is correct because the integral is solved using substitution and properties of definite integrals.

773 Medium

773 of 995

Application +4 | -1

\int \nolimits_0^{\pi /4} \,\,\log \,\,(1 + \tan \,\,x)\,\,dx

is equal to

View answer and explanation

Correct option: A

\frac{\pi }{8}\,\,{\log _e}\,\,2

Option A is correct because the integral is solved by using properties of definite integrals and logarithmic functions.

774 Medium

774 of 995

Application +4 | -1

I = \int \limits_0^1 \,\,\cos \,\,\left\{ {2{{\cot }^{ - 1}}\,\,\sqrt {\frac{{1 - x}}{{1 + x}}} } \right\}\,\,dx

, then

View answer and explanation

Correct option: B

I = - \,\,\frac{1}{2}

Option C (

\pi

) is correct because the integral of the odd functions over the symmetric interval is zero, and only the integral of 1 remains.

775 Easy

775 of 995

Analyzing +4 | -1

Let

f

be a real valued continuous function on

[0,1]

and

f(x)=x+\int_0^1(x-t) f(t) d t

. Then, which of the following points

(x, y)

lies on the curve

y=f(x)

View answer and explanation

Correct option: D

(6,8)

Option D ((6,8)) is correct because substituting x=6 into the derived function f(x) yields y=8, satisfying the equation.

776 Medium

776 of 995

Application +4 | -1

\int\limits_{0}^{2 a} \frac{f(x)}{f(x)+f(2 a-x)} d x=

View answer and explanation

Correct option: B

a

Option B is correct because the integral can be solved using the property

\int_a^b \frac{f(x)}{f(x)+f(a+b-x)} dx = \frac{b-a}{2}

777 Easy

777 of 995

Application +4 | -1

The value of

\int\limits_0^{\pi /2} {\frac{{{{\sin }^3}x}}{{\sin x + \cos x}}dx} \,

View answer and explanation

Correct option: C

\frac{{\pi - 1}}{4}

Option C (

\frac{{\pi - 1}}{4}

) is correct because, by using the property of definite integrals and simplifying, the integral can be evaluated to this value.

778 Medium

778 of 995

Application +4 | -1

\int \nolimits_{\pi /6}^{\pi /3} \,\,\frac{{dx}}{{1 + \sqrt {\cot \,\,x} }}

is equal to

View answer and explanation

Correct option: C

\frac{\mathrm{\pi }}{12}

Option C (

\frac{\mathrm{\pi }}{{12}}

) is correct because the definite integral is evaluated using the property

\int \nolimits_a^b \,f\,\,\left( x \right)\,\,dx = \int \nolimits_a^b \,f\,\,\left( {a + b - x} \right)\,\,dx

779 Easy

779 of 995

Application +4 | -1

\int\limits_{0}^a \frac{d x}{x+\sqrt{a^2-x^2}}=

View answer and explanation

Correct option: B

\frac{\pi}{4}

Option B (

\frac{\pi}{4}

) is correct because the integral evaluates to

\frac{\pi}{4}

after substitution and simplification.

780 Easy

780 of 995

Understanding +4 | -1

The value of

\int \nolimits_{ - 2}^2 \,\,\left( {a{x^3}\, + \,\,bx\,\, + \,\,c} \right)dx

depends on the

View answer and explanation

Correct option: B

Value of

c

Option B (Value of

c

) is correct because the integral of odd functions over a symmetric interval is zero, leaving only the constant term's integral to depend on

c

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