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

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

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

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

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

If $\mathop \smallint \limits_{ - 1}^4 \,\,f\left( x \right)\,\,dx = 4$ and $\mathop \smallint \limits_2^4 \,\,\left( {3 - f\left( x \right)} \right)\,\,dx = 7$, then the value of $\mathop \smallint \limits_2^{ - 1} \,\,f\left( x \right)dx$, is

$\begin{array}{*{20}{l}} {\rm{2}} \end{array}$

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JEE • MATHEMATICS • DEFINITE INTEGRATION

Definite integration practice · Easy

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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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Showing 1-30 of 791 questions

1 Easy

1 of 791

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 Easy

2 of 791

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.

3 Easy

3 of 791

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.

4 Easy

4 of 791

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.

5 Easy

5 of 791

Application +4 | -1

Area bounded by

y=\sec ^2 x, x=\frac{\pi}{6},

x=\frac{\pi}{3}

and x-axis is

View answer and explanation

Correct option: A

\frac{2}{\sqrt{3}}

Option A (

\frac{{2}}{{\sqrt{{3}}}}

) is correct because it is the result of integrating

\sec^2 x

from

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

\frac{{\pi}}{{3}}

6 Easy

6 of 791

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

7 Easy

7 of 791

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.

8 Easy

8 of 791

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.

9 Easy

9 of 791

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

10 Easy

10 of 791

Application +4 | -1

Area bounded by curve

y=x^3, x

-axis and ordinates

x=1

and

x=4

, is

View answer and explanation

Correct option: D

\frac{255}{4}

sq. unit

Option D (

\frac{{255}}{{4}}

sq. unit) is correct because the area is calculated by integrating

x^3

from

x=1

x=4

11 Easy

11 of 791

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}

12 Easy

12 of 791

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.

13 Easy

13 of 791

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.

14 Easy

14 of 791

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.

15 Easy

15 of 791

Application +4 | -1

Given

f(x)=\left\{\begin{array}{cc} x+1 & ; x<0 \\ \cos \pi x & ; x \geq 0 \end{array}\right.

then the value of

\int_{-1}^1 f(x) d x

is equal to

View answer and explanation

Correct option: C

\frac{1}{2}

Option C (

\frac{1}{2}

) is correct because it involves splitting the integral based on the piecewise function definition and evaluating each part separately.

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 791

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.

17 Easy

17 of 791

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.

18 Easy

18 of 791

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}

19 Easy

19 of 791

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.

20 Easy

20 of 791

Application +4 | -1

The area of figure bounded by

y=e^x, y=e^{-x}

and the straight line

x=1

View answer and explanation

Correct option: C

e+\frac{1}{e}-2

Option C (

e+\frac{1}{e}-2

) is correct because it represents the definite integral of the difference between the two exponential functions from 0 to 1.

21 Easy

21 of 791

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}

22 Easy

22 of 791

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.

23 Easy

23 of 791

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.

24 Easy

24 of 791

Analyzing +4 | -1

The integral

\int\limits_{{\pi / 6}}^{\pi / 4} \frac{\mathrm{d} x}{\sin 2 x\left(\tan ^5 x+\cot ^5 x\right)}

equals :

View answer and explanation

Correct option: D

\frac{1}{5}\left(\frac{\pi}{4}-\tan ^{-1}\left(\frac{1}{3 \sqrt{3}}\right)\right)

Option D (

\frac{1}{5}(\frac{\pi}{4}-\tan ^{-1}(\frac{1}{3 \sqrt{3}}))

) is correct because the integral is solved by using trigonometric substitution and simplification.

25 Easy

25 of 791

Application +4 | -1

\int \limits_{ - 1}^4 \,\,f\left( x \right)\,\,dx = 4

and

\int \limits_2^4 \,\,\left( {3 - f\left( x \right)} \right)\,\,dx = 7

, then the value of

\int \limits_2^{ - 1} \,\,f\left( x \right)dx

, is

View answer and explanation

Correct option: C

- \,5

Option C (-5) is correct because the integral can be evaluated using the properties of definite integrals and the given information.

26 Easy

26 of 791

Application +4 | -1

\int \nolimits_0^1 \,\,\frac{{{x^2}}}{{1 + {x^2}}}\,\,dx

is equal to

View answer and explanation

Correct option: D

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

Option D (

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

) is correct because the definite integral is evaluated by rewriting the integrand and using the arctangent integral.

27 Easy

27 of 791

Understanding +4 | -1

Find the equation of the tangent of the tangent to the curve

y=\int_{x^2}^{x^2} \frac{d t}{\sqrt{1+t^2}}

x=1

View answer and explanation

Correct option: C

y=\frac{1}{\sqrt{2}}(x-1)

Option C (

y=\frac{{1}}{{\sqrt{{2}}}}(x-1)

) is correct because it represents the equation of the tangent line to the curve at x=1, found by evaluating the derivative at that point.

28 Easy

28 of 791

Analyzing +4 | -1

\lim \limits_{{n \rightarrow \infty}}\left\{\frac{1}{n}+\frac{n}{(n+1)^2}+\frac{n}{(n+2)^2}+\ldots+\frac{n}{(2 n-1)^2}\right\}

is equal to

View answer and explanation

Correct option: A

\frac{1}{2}

Option B (

e^{-1}

) is correct because the limit can be transformed into an integral using logarithms and Riemann sums, which evaluates to

e^{-1}

29 Easy

29 of 791

Application +4 | -1

Let

p(x)

be a function defined on

\mathbf{R}

such that

p^{\prime}(x)=p^{\prime}(1-x)

, for all

x \in[0,1], p(0)=1

and

p(1)=41

.
Then

\int\limits_{0}^1 p(x) d x

equals

View answer and explanation

Correct option: D

Option D is correct because the integral is solved by using integration by parts and the given property of the function p(x).

30 Easy

30 of 791

Application +4 | -1

{I_1} = \int\limits_0^1 {{{(1 - {x^{50}})}^{100}}} dx

and

{I_2} = \int\limits_0^1 {{{(1 - {x^{50}})}^{101}}} dx

such that

{I_2} = \alpha {I_1},

then

\mathrm{\alpha }

equals to

View answer and explanation

Correct option: C

\frac{{5050}}{{5051}}

Option D (

\frac{{5050}}{{5051}}

) is correct because it represents the ratio between the two integrals, derived using integration by parts.

Revision notes and key takeaways

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

Key Takeaways

✓ This set contains 791 single-correct-answer MCQs on DEFINITE INTEGRATION.
✓ Try each question before opening its answer. Read the explanation where one is available.
✓ Sign in to save submitted answers. Revisit incorrect answers alongside your textbook or class notes.

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