The value $\left( {\rm{s}} \right)$ of $\mathop \smallint \nolimits_0^1 \,\,\frac{{{x^4}{{\left( {1 - x} \right)}^4}}}{{1 + {x^2}}}\,\,dx$ is (are)

$\frac{{71}}{{15}}\,\, - \,\,\frac{{3\,\pi }}{2}$

Let $F\left( x \right) = f\left( x \right)\,\, + \,\,f\left( {\frac{1}{x}} \right)$, where $f\left( x \right) = \mathop \smallint \nolimits_1^x \,\,\frac{{\log \,\,t}}{{1\,\, + \,\,t}}\,\,dt.\,\,{\rm{Then}},\;\,\,F\left( e \right)\,\,{\rm{equals}}$

$\begin{array}{*{20}{l}} {1/2} \end{array}$

If $f(a+b-x)=f(x)$ then $\int_\limits a^b x f(x) d x$ is equal to

$\frac{a+b}{2} \int_\limits a^b f(a+b+x) d x$

$\frac{a+b}{2} \int_\limits a^b f(b-x) d x$

$\frac{b-a^b}{2} \int_\limits a^b f(x) d x$.

If \[f\,\,\left( {a + b - x} \right) = f\,\,\left( x \right),\,\,\;{\rm{then}}\,\,\mathop \smallint \nolimits_a^b \,\,xf\,\,\left( x \right)\,\,dx\] is equal to

$\frac{{a + b}}{2}\,\,\mathop \smallint \nolimits_a^b \,\,f\left( x \right)\,\,dx$

$\frac{{a + b}}{2}\,\,\mathop \smallint \nolimits_a^b \,\,f\left( {b - x} \right)\,\,dx$

$\frac{{b - a}}{2}\,\,\mathop \smallint \nolimits_a^b \,\,f\left( x \right)\,\,dx$

$\frac{{a + b}}{2}\,\,\mathop \smallint \nolimits_a^b \,\,f\left( {a + b + x} \right)\,\,dx$

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JEE • MATHEMATICS • DEFINITE INTEGRATION

Definite integration practice · Mixed difficulty

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Practice format and chapter quiz

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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Showing 1-30 of 995 questions

1 Easy

1 of 995

Analyzing +4 | -1

\int \nolimits_0^\infty \,\,\log \,\,\left( {x + \frac{1}{x}} \right)\,\,\frac{{dx}}{{1 + {x^2}}}

is equal to

View answer and explanation

Correct option: A

\mathrm{\pi }\,\,\log \,\,2

The correct answer is C (

\pi /2

) because the integral can be solved by splitting the integrand and using integration by parts.

2 Medium

2 of 995

Application +4 | -1

f\left( x \right)

is differentiable and

\int \nolimits_0^{{t^2}} \,\,x\;\,f\left( x \right)\,\,dx = \frac{2}{5}\,\,{t^5}

, then

f\left( {\frac{4}{{25}}} \right)

equals

View answer and explanation

Correct option: A

\frac{2}{5}

Option A (2/5) is correct because after differentiating both sides of the given equation using the Newton-Leibniz rule and solving for f(x), we find f(4/25) = 2/5.

3 Easy

3 of 995

Application +4 | -1

The definite integral

\int\limits_{{-2}}^0\left(x^3+3 x^2+3 x+3+(x+1) \cos (x+1)\right) d x

equals

View answer and explanation

Correct option: C

Option C (4) is correct because simplifying the integral by substitution and recognizing odd functions results in a straightforward calculation.

4 Medium

4 of 995

Analyzing +4 | -1

I\,\,\left( {m,\,\,n} \right) = \int \nolimits_0^1 \,\,{t^m}\,\,{\left( {1 + t} \right)^n}\,\,dt

,then the expression for

I\left( {m,\,\,n} \right)

in terms of

I\,\,\left( {m + 1,\,\,n + 1} \right)

View answer and explanation

Correct option: A

\frac{{{2^n}}}{{m + 1}}\,\, - \,\,\,\frac{n}{{m + 1}}\,\,\left( {m + 1,n - 1} \right)

Option A (

\frac{{{2^n}}}{{m + 1}}\, - \,\,\,\frac{n}{{m + 1}}\,\,\,\left( {m + 1,n - 1} \right)

) is correct because integration by parts leads to this recursive relationship.

5 Medium

5 of 995

Analyzing +4 | -1

I = \int \limits_3^4 \,\,\frac{1}{{\sqrt[3]{{\log \,\,x}}}}\,\,dx

, then

View answer and explanation

Correct option: A

0.92 < I < 1

Option A (

0.92 < I < 1

) is correct because the integral is bounded by comparing the integrand with simpler functions.

6 Easy

6 of 995

Application +4 | -1

The value of integration

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

View answer and explanation

Correct option: C

4 \ln \left(\frac{4}{3}\right)

Option C (

4 \ln \left(\frac{4}{3}\right)

) is correct because simplifying the integrand and evaluating the definite integral results in this value.

7 Easy

7 of 995

Application +4 | -1

J_m=\int\limits_{1}^e \log ^m x \mathrm{~d} x

then

J_8+8 J_7

is equal to

View answer and explanation

Correct option: D

e

Option D (e) is correct because using integration by parts, J_8 + 8J_7 simplifies to e.

8 Medium

8 of 995

Analyzing +4 | -1

\lim\limits_{n \to \infty } \,\,{\left\{ {\left( {1 + \,\,\frac{1}{n}} \right)\,\,\left( {1 + \,\,\frac{2}{n}} \right)\,\,\left( {1 + \,\,\frac{3}{n}} \right)\,\, \ldots \,\,\left( {1 + \,\,\frac{n}{n}} \right)} \right\}^{1/n}}

is equal to

View answer and explanation

Correct option: B

4/e

Option B (

4/e

) is correct because taking the logarithm and converting the sum to an integral allows us to evaluate the limit.

9 Easy

9 of 995

Application +4 | -1

Q(x)=\frac{1}{2} a_0+a_1 \cos x+b_1 \sin x+a_2 \cos 2 x

+\ldots+a_n \cos n x+b_n \sin n x

, then the value of

\int\limits_{0}^{2 \pi} Q(x) \sin k x \mathrm{~d} x(k=1,2, \ldots n)

View answer and explanation

Correct option: D

\pi b_k

Option D (

\pi b_k

) is correct because when integrating

Q(x) \sin kx

, only the term

b_k \sin kx \sin kx

contributes a non-zero value, resulting in

\pi b_k

10 Easy

10 of 995

Application +4 | -1

\mathrm{g}\left( x \right) = \frac{{f\left( x \right) - f\left( { - x} \right)}}{2}

defined over

\left[ { - \,3,\,\,3} \right]

and

f\left( x \right) = 2{x^2} - 4x + 1

, then

\int \nolimits_{ - 3}^3 \,\,\,\mathrm{g}\,\,\left( x \right)\,\,dx

is equal to

View answer and explanation

Correct option: A

\begin{array}{llllllllllllllllllll} \mathrm{0} \end{array}

Option A is correct because the integral of the odd function g(x) over the symmetric interval [-3, 3] is 0.

11 Easy

11 of 995

Application +4 | -1

For any natural number

n

, the value of

\int_0^{n^2}[\sqrt{x}], d x

View answer and explanation

Correct option: B

\frac{n(n-1)(4 n+1)}{6}

Option B (

\frac{n(n-1)(4 n+1)}{6}

) is correct because it represents the sum of the areas under the greatest integer function of the square root of x.

12 Easy

12 of 995

Analyzing +4 | -1

The value

\left( \mathrm{s} \right)

\int \nolimits_0^1 \,\,\frac{{{x^4}{{\left( {1 - x} \right)}^4}}}{{1 + {x^2}}}\,\,dx

is (are)

View answer and explanation

Correct option: A

\frac{{22}}{7}\,\, - \,\,\pi

Option A

\frac{{22}}{{7}}\, - \,\pi

is correct because after expanding and integrating the given expression, the result simplifies to

\frac{22}{7} - \pi

13 Easy

13 of 995

Application +4 | -1

\int \nolimits_0^p \,\,\frac{{dx}}{{1\,\, + \,\,4{x^2}}} = \frac{\pi }{8}

, then the value of

p

View answer and explanation

Correct option: D

\frac{1}{2}

Option D (

\frac{1}{2}

) is correct because solving the definite integral and equating it to

\frac{\pi}{8}

yields

p = \frac{1}{2}

14 Medium

14 of 995

Application +4 | -1

The value of

\int_{-1}^2|[x]-\{x\}| d x

, where

[x]

is the greatest integer less than or equal to

x

and

\{x\}

is the fractional part of

x

, is

View answer and explanation

Correct option: B

5 / 2

Option B (5/2) is correct because it is the result of integrating the absolute value of the difference between the greatest integer function and the fractional part function over the given interval.

15 Medium

15 of 995

Understanding +4 | -1

f\left( t \right)

is a continuous function defined on

\left[ {a,\;\,b} \right]

such that

f\left( t \right)

is an odd function, then the function

\phi \,\,\left( x \right) = \int \limits_a^x \,\,f\left( t \right)dt

View answer and explanation

Correct option: B

Is an even function

Option B (Is an even function) is correct because when f(t) is an odd function, its integral from a to x results in an even function.

Explore Other Chapters in MATHEMATICS

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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

16 Easy

16 of 995

Application +4 | -1

The integral

\int_{-1 / 2}^{1 / 2}\left\{[x]+\ln \left(\frac{1+x}{1-x}\right)\right\} d x

equals

View answer and explanation

Correct option: A

-\frac{1}{2}

Option A (

-\frac{1}{2}

) is correct because it is the value of the integral after evaluating the greatest integer function and recognizing the odd function.

17 Easy

17 of 995

Application +4 | -1

Let

F\left( x \right) = f\left( x \right)\,\, + \,\,f\left( {\frac{1}{x}} \right)

, where

f\left( x \right) = \int \nolimits_1^x \,\,\frac{{\log \,\,t}}{{1\,\, + \,\,t}}\,\,dt.\,\,\mathrm{Then},\;\,\,F\left( e \right)\,\,\mathrm{equals}

View answer and explanation

Correct option: A

\begin{array}{llllllllllllllllllll} {1/2} \end{array}

Option A (1/2) is correct because the integral F(e) simplifies to 1/2 after applying the given function definition and substitution.

18 Easy

18 of 995

Analyzing +4 | -1

Suppose that

f\,\,\left( x \right)\,\,\, \ge \,\,0

for all

x\,\, \in \,\,\left[ {0,1} \right]

and

f

is continuous in

\left[ {\mathrm{0,}\,\mathrm{1}} \right]

and

\int \nolimits_0^1 \,\,f\left( x \right)\,\,dx = 0

then

\forall \,\,x\,\, \in \,\,\left[ {0,1} \right],\,\,f

View answer and explanation

Correct option: C

Constant

Option C (Constant) is correct because if f(x) is non-negative and its integral over [0,1] is 0, then f(x) must be identically zero, hence constant.

19 Medium

19 of 995

Application +4 | -1

Let

f(x)=\int\limits_{x}^{x^2}(t-1) d t

. Then the value of

\left|f^{\prime}(\omega)\right|

where

\omega

is a complex cube root of unity is

View answer and explanation

Correct option: C

3 \sqrt{3}

Option C (

3 \sqrt{3}

) is correct because it is the absolute value of the derivative of the function evaluated at a complex cube root of unity.

20 Easy

20 of 995

Application +4 | -1

If the value of

\int\limits_{{-2}}^2|x \cos \pi x| \mathrm{d} x=k / \pi

then the value of

k

View answer and explanation

Correct option: B

Option B (8) is correct because the integral evaluates to 8/

\pi

, thus k=8.

21 Easy

21 of 995

Application +4 | -1

\int\limits_{0}^{10 \pi}|\sin x| d x

View answer and explanation

Correct option: A

Option A (

\frac{1}{5}

) is correct because the limits are converted to definite integrals and evaluated.

22 Easy

22 of 995

Application +4 | -1

\int\limits_{2}^3 \frac{\sqrt{x}}{\sqrt{5-x}+\sqrt{x}} d x=

View answer and explanation

Correct option: B

\frac{1}{2}

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}

23 Easy

23 of 995

Application +4 | -1

f(a+b-x)=f(x)

then

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

is equal to

View answer and explanation

Correct option: C

\frac{a+b}{2} \int\limits_{a}^b f(x) d x

Option C (

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

) is correct because using the given property simplifies the integral.

24 Medium

24 of 995

Application +4 | -1

The value of the integral

\int \limits_1^e \,{\left( {\log \,\,x} \right)^3}dx

View answer and explanation

Correct option: B

6\,\, - \,\,2e

Option C (

2e\\ - \\ 6

) is correct because the integral is solved using integration by parts repeatedly.

25 Medium

25 of 995

Application +4 | -1

\int\limits_{1}^{e^3} \frac{d x}{x \sqrt{1+\log _e x}}=

View answer and explanation

Correct option: A

Option A is correct because by substituting

\log_e x = t

, the integral simplifies to

\int_0^3 \frac{dt}{\sqrt{1+t}}

, which evaluates to 2.

26 Easy

26 of 995

Application +4 | -1

\int \nolimits_0^{{x^2}} \,\,f\left( t \right)\;dt = x\,\,\cos \,\,\pi x

, then the value of

f\left( 4 \right)

View answer and explanation

Correct option: B

\frac{1}{4}

Option B (

\frac{1}{4}

) is correct because after differentiating both sides of the given equation and solving for

f(x^2)

, we find

f(4) = \frac{1}{4}

27 Easy

27 of 995

Application +4 | -1

\int \limits_\pi ^{10\pi } \,\,\left| {\,\sin \,\,x\,} \right|\,\,dx

is equal to

View answer and explanation

Correct option: D

\mathrm{18}

Option D (

\mathrm{18}

) is correct because the integral of the absolute value of sine over multiple periods simplifies to a multiple of the integral over one period.

28 Medium

28 of 995

Application +4 | -1

The value of the integral

\int \limits_0^{\pi /4} \,\,\frac{{\sin \,\,\theta \,\, + \,\,\cos \,\,\theta }}{{9\,\, + \,\,16\,\,\sin \,\,2\theta }}\,\,d\theta

, is

View answer and explanation

Correct option: C

\frac{1}{{20}}\,\,\log \,\,3

Option C (

\frac{1}{{20}}\\\,\log \\,3

) is correct because it is the value obtained after evaluating the definite integral using substitution.

29 Easy

29 of 995

Application +4 | -1

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

is equal to

View answer and explanation

Correct option: B

\frac{{a + b}}{2}\,\,\int \nolimits_a^b \,\,f\left( x \right)\,\,dx

Option B (

\frac{{a + b}}{2}\,\int \nolimits_a^b \,f\left( x \right)\,dx

) is correct because it uses the property of definite integrals with the given functional relationship to simplify the integral.

30 Medium

30 of 995

Application +4 | -1

The area of the region bounded by

y=x^2

and

y=-x^2+2

View answer and explanation

Correct option: C

\frac{8}{3}

Option C (8/3) is correct because it represents the area calculated by integrating the difference between the two parabolas from their intersection points.

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Key Takeaways

✓ This set contains 995 single-correct-answer MCQs on DEFINITE INTEGRATION.
✓ Try each question before opening its answer. Read the explanation where one is available.
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