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

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

The difference between the greatest and least values of the function $\phi \,\,\left( x \right) = \mathop \smallint \limits_0^x \left( {t + 1} \right)\,\,dt$ on $\left[ {{\rm{2, 3}}} \right]$, is

${\rm{11}}\,{\rm{/}}\,{\rm{2}}$

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

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

$\frac{7}{2}\,\, - \,\,2{\pi ^3}$

$\mathop \smallint \nolimits_8^{15} \,\,\frac{{dx}}{{\left( {x - 3} \right)\,\,\sqrt {x + 1} }}$ is equal to

$\frac{1}{3}\,\,\log \,\,\frac{5}{3}$

$\frac{1}{5}\,\,\log \,\,\frac{3}{5}$

$\frac{1}{2}\,\,\log \,\,\frac{3}{5}$

${I_{m,n}} = \mathop {\smallint \,\,}\limits_0^1 {x^m}\,\,{\left( {\ln \,\,x} \right)^n}\,\,dx = $

$\frac{n}{{n + 1}}\,\,{I_{m,\,n - 1}}$

$\frac{{ - \,m}}{{n + 1}}\,\,{I_{m,n - 1}}$

$\frac{m}{{n + 1}}\,\,{I_{m,n - 1}}$

Back to DEFINITE INTEGRATION

Your Progress

0 / 995 Answered

Correct

Incorrect

Accuracy

JEE • MATHEMATICS • DEFINITE INTEGRATION

Definite integration practice · Mixed difficulty

Choose an answer to check it, or open “View answer” to study the explanation. Sign in to save progress to your account.

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.

Take a chapter quiz →

Showing 31-60 of 995 questions

Preparing practice and saved progress…

31 Easy

31 of 995

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.

32 Easy

32 of 995

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.

33 Medium

33 of 995

Application +4 | -1

The value of

k

for which the area of the figure bounded by the curve

y = 8\;{x^2} - {x^5},

the straight line

x = 1

and

x = k

and the

x

-axis is equal to 16/3

View answer and explanation

Correct option: B

\sqrt[3]{{8 - \sqrt {17} }}

The correct answer is B (

\sqrt[3]{{8 - \sqrt {17} }}

) because it is obtained by solving the definite integral of the given function between the limits and equating it to the given area.

34 Easy

34 of 995

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.

35 Medium

35 of 995

Application +4 | -1

Let

f\left( x \right)

be a function satisfying

f'\,\,\left( x \right) = f\,\,\left( x \right)

with

f\left( 0 \right) = 1

and

g\left( x \right)

be the function satisfying

f\left( x \right)\,\, + \,\,g\left( x \right) = {x^2}

. The, value of the integral

\int \limits_0^1 \,f\left( x \right)\,\,g\,\,\left( x \right)dx

, is

View answer and explanation

Correct option: D

None of these

Option D (None of these) is correct because the calculated integral value does not match any of the provided options. The correct integral is

e - \frac{1}{2}{e^2} - \frac{3}{2}

36 Easy

36 of 995

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).

37 Easy

37 of 995

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.

38 Easy

38 of 995

Application +4 | -1

The difference between the greatest and least values of the function

\phi \,\,\left( x \right) = \int \limits_0^x \left( {t + 1} \right)\,\,dt

\left[ {\mathrm{2, 3}} \right]

, is

View answer and explanation

Correct option: C

\mathrm{7}\,\mathrm{/}\,\mathrm{2}

Option C is correct because the difference between the greatest and least values is found by evaluating the definite integral at the bounds and subtracting.

39 Easy

39 of 995

Application +4 | -1

f(x)=\int\limits_{0}^x \sin ^8 t \mathrm{~d} t

, then

f(x+\pi)

equals

View answer and explanation

Correct option: C

f(x)+f(\pi)

Option C (

f(x)+f(\pi)

) is correct because

f(x+\pi)

can be split into two integrals, one from 0 to

\pi

and the other from

\pi

x+\pi

, which simplifies to

f(x)+f(\pi)

40 Medium

40 of 995

Application +4 | -1

The value of the integral

\int\limits_{{1 / 2}}^2 \frac{\tan ^{-1} x}{x} d x

is equal to

View answer and explanation

Correct option: C

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

Option C (

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

) is correct because by using the property of definite integrals and substituting

x

with

\frac{1}{x}

, we can simplify the integral and solve it.

41 Medium

41 of 995

Application +4 | -1

Assume that

f

is continuous everywhere, then

\frac{1}{c} \int\limits_{x}^{b c} f\left(\frac{x}{c}\right) d x=

View answer and explanation

Correct option: C

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

Option C (

\int_a^b f(x) dx

) is correct because it is the result of applying a u-substitution to the integral, which simplifies the expression to the definite integral of f(x) from a to b.

42 Medium

42 of 995

Application +4 | -1

The set of values of '

a

' which satisfy the equation

\int_0^2\left(t-\log _2 a\right) d t=\log _2\left(\frac{4}{a^2}\right)

, is

View answer and explanation

Correct option: B

a \in R^{+}

Option B is correct because the equation simplifies to an identity, implying that the solution set is all positive real numbers.

43 Medium

43 of 995

Application +4 | -1

The area of the figure bounded by

y = \sin x,y = \cos x

in the first quadrant, is

View answer and explanation

Correct option: A

2\left( {\sqrt 2 - 1} \right)

Option A (

2\left( {\sqrt 2 - 1} \right)

) is correct because it represents the sum of the definite integrals of

(\cos x - \sin x)

from 0 to

\frac{\pi}{4}

and

(\sin x - \cos x)

from

\frac{\pi}{4}

\frac{\pi}{2}

44 Easy

44 of 995

Application +4 | -1

\int \nolimits_0^\mathrm{\pi } \,\,\frac{{\mathrm{\theta }\,\,\sin \,\,\mathrm{\theta }}}{{1\,\, + \,\,{{\cos }^2}\,\,\mathrm{\theta }}}\,\,\mathrm{d\theta }

is equal to

View answer and explanation

Correct option: D

\frac{{{\pi ^2}}}{4}

Option A (

\frac{{{\pi ^2}}}{4}

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

45 Easy

45 of 995

Understanding +4 | -1

The value of

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

View answer and explanation

Correct option: A

\mathrm{0}

Option A (0) is correct because the integrand is an odd function, and the integral of an odd function over a symmetric interval from

-\pi

\pi

is zero.

Explore Other Chapters in MATHEMATICS

View All →

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

46 Easy

46 of 995

Application +4 | -1

\int\limits_{0}^2\left[x^2\right] d x

View answer and explanation

Correct option: D

-\sqrt{2}-\sqrt{3}+5

Option B (

e-1

) is correct because the limit of the sum is converted to a definite integral and evaluated.

47 Easy

47 of 995

Application +4 | -1

The value of the integral

\int \limits_\alpha ^\beta \,\,\frac{1}{{\sqrt {\left( {x - \alpha } \right)\,\,\left( {\beta - x} \right)} }}\,\,dx

for

\beta \,\,\, > \,\,\alpha

, is

View answer and explanation

Correct option: D

\pi

Option D (

\pi

) is correct because the integral evaluates to

\pi

after completing the square and using trigonometric substitution.

48 Easy

48 of 995

Application +4 | -1

The value of the integral

\int\limits_{{-\pi / 3}}^{\pi / 3} \frac{x \sin x}{\cos ^2 x} \mathrm{~d} x

View answer and explanation

Correct option: B

2\left(\frac{2 \pi}{3}-\log (2+\sqrt{3})\right)

The correct answer is

2 \sqrt{{2}} a

because the sum of the intercepts forms a geometric progression with a common ratio of 1/2, and the sum to infinity is calculated.

49 Medium

49 of 995

Application +4 | -1

\lim\limits_{x \to \frac{\pi }{4}} \frac{{\int\limits_2^{{{\sec }^2}x} {f\left( t \right)dt} }}{{{x^2} - \frac{{{\pi ^2}}}{{16}}}}

equals

View answer and explanation

Correct option: A

\frac{8}{\pi} f(2)

Option A (

\frac{8}{\pi} f(2)

) is correct because it is the result of applying L'Hopital's rule to the limit and evaluating at

x = \frac{\pi}{4}

50 Easy

50 of 995

Analyzing +4 | -1

\lim\limits_{n \to \infty } \left( {\frac{n}{{{n^2} + {1^2}}} + \frac{n}{{{n^2} + {2^2}}} + \frac{n}{{{n^2} + {3^2}}} + ... + \frac{1}{{5n}}} \right)

is equal to

View answer and explanation

Correct option: B

{\tan ^{ - 1}}\left( 2 \right)

Option B (

{\tan ^{ - 1}}( 2 )

) is correct because the limit of the given summation can be expressed as a definite integral, which evaluates to

{\tan ^{ - 1}}(2)

51 Easy

51 of 995

Application +4 | -1

The area enclosed by the curve

\frac{{{x^2}}}{{25}} + \frac{{{y^2}}}{{16}} = 1

View answer and explanation

Correct option: B

20\pi

sq unit

Option B,

20\pi

sq unit, is the correct area of the ellipse because the area of an ellipse is given by

\pi ab

, where a and b are the semi-major and semi-minor axes, respectively.

52 Medium

52 of 995

Analyzing +4 | -1

f(x) = \left| \begin{array}{l} \sin \,\,x\,\, + \,\,\sin \,\,\,2x\,\, + \,\,\sin \,\,\,3x\,\,\,\sin \,\,\,2x\,\,\,\,\,\,\,\,\,\,\sin \,\,3x\\ \,\,\,\,\,3 + 4\sin x\;\,\,\,\,\,\,\,\,\,\;\;\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,3\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,4\,\,\sin \,\,x\\ 1\,\, + \,\,\sin \,\,x\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\sin \,\,x\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;1\;\;\;\; \end{array} \right|\,

, then the value of

\int \nolimits_0^{\pi /2} \,\,f\left( x \right)\,\,dx

View answer and explanation

Correct option: C

\frac{1}{3}

Option C (

\frac{1}{3}

) is correct because the determinant simplifies to

\sin 3x

, and the integral of

\sin 3x

from 0 to

\pi/2

\frac{1}{3}

53 Medium

53 of 995

Analyzing +4 | -1

If for

n \geq 1, P_n=\int\limits_{1}^e(\log x)^n d x

, then

P_{10}-90 P_8

is equal to:

View answer and explanation

Correct option: C

-9 \mathrm{e}

Option C (-9e) is correct because it is the result of applying integration by parts and simplifying the expression.

54 Medium

54 of 995

Application +4 | -1

If for a continuous function

f(x), \int\limits_{{-\pi}}^t(f(x)+x) d x=\pi^2-

t^2

, for all

t \geq-\pi

, then

f\left(-\frac{\pi}{3}\right)

is equal to:

View answer and explanation

Correct option: A

\pi

Option A (

\pi

) is correct because after differentiating the given integral equation and solving for f(x), we find f(-

\frac{\pi}{3}

) =

\pi

55 Easy

55 of 995

Application +4 | -1

For

0 < \alpha < \pi

. The value of the definite integral

\int\limits_{0}^1 \mathrm{~d} x /\left(x^2+2 x \cos \alpha+1\right)

is equal to

View answer and explanation

Correct option: A

\alpha /(2 \sin \alpha)

Option A (

\alpha /(2 \sin \alpha)

) is correct. The integral is solved by completing the square, using a trigonometric substitution, and simplifying the resulting expression using trigonometric identities.

56 Easy

56 of 995

Application +4 | -1

\int \nolimits_8^{15} \,\,\frac{{dx}}{{\left( {x - 3} \right)\,\,\sqrt {x + 1} }}

is equal to

View answer and explanation

Correct option: A

\frac{1}{2}\,\,\log \,\,\frac{5}{3}

Option A (

\frac{1}{2}\,\log \,\frac{5}{3}

) is correct because after substituting

x + 1 = t^2

and evaluating the definite integral, the result is

\frac{1}{2} \log \frac{5}{3}

57 Easy

57 of 995

Application +4 | -1

\int \nolimits_0^\pi \,\,x\,\,\mathrm{ }f\,\,\left( {\sin \,\,x} \right)\,\,dx = A\,\,\int \nolimits_0^{\pi /2} \,\,f\,\,(\sin \,\,x)\,\,dx,\mathrm{then}\,\,A

is equal to

View answer and explanation

Correct option: B

\pi

Option B (

\pi

) is correct because by using the property of definite integrals, the given integral can be expressed in terms of the integral from 0 to

\pi/2

, and A is found to be

\pi

58 Medium

58 of 995

Application +4 | -1

If the area above

x

-axis, bounded by the curves

y = {2^{kx}}

and

x = 0

and

x = 2

\frac{3}{{\log 2}},

then the value of

k

View answer and explanation

Correct option: B

Option B (1) is correct because substituting k = 1 into the area equation derived from the integral satisfies the given condition.

59 Medium

59 of 995

Application +4 | -1

{I_{m,n}} = \mathop {\int \,\,}\limits_0^1 {x^m}\,\,{\left( {\ln \,\,x} \right)^n}\,\,dx =

View answer and explanation

Correct option: C

\frac{{ - \,\,n}}{{m + 1}}\,\,{I_{m,n - 1}}

Option C (

\frac{{ - \\,n}}{{m + 1}}\\,,{I_{m,n - 1}}

) is correct because it is the result of applying integration by parts to the given integral.

60 Easy

60 of 995

Application +4 | -1

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

View answer and explanation

Correct option: B

4/15

Option B (

4/15

) is correct. The integral is evaluated by splitting it into odd and even functions and using trigonometric identities.

Revision notes and key takeaways

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

Showing 31-60 of 995 questions

Your Progress

Definite integration practice · Mixed difficulty

Explore Other Chapters in MATHEMATICS

Continue Your Practice

Your practice