$\mathop \smallint \nolimits_{ - 10}^{10} \,\,\log \,\,\left( {\frac{{a\,\, + \,\,x}}{{a\,\, - \,\,x}}} \right)\,\,dx$ is equal to

$ - \,2\,\,\log \,\,\left( {a + 10} \right)$

$2\,\,\log \,\,\left( {\frac{{a\,\, + \,\,10}}{{a\,\, - \,\,10}}} \right)$

$2\,\,\log \,\,\left( {a\,\, + \,\,10} \right)$

The value of $\smallint {e^{2x}}\,(2\,\,\sin \,\,3x\,\, + \,\,3\,\,\cos \,\,3x)\,dx$ is

${e^{2x}}\,\,\cos \,\,3x\,\, + \,\,c$

${e^{2x}}\;(2\,\,\sin \,\,3x)\,\, + \,\,c$

If $I\, = \,\int_1^2 {\frac{{dx}}{{\sqrt {2{x^3} - 9{x^2} + 12x + 4} }},} $ then

$\frac{1}{6} < {I^2} < \frac{1}{2}$

$\frac{1}{8} < {I^2} < \frac{1}{4}$

$\frac{1}{{16}} < {I^2} < \frac{1}{9}$

If $f\,\left( x \right)$ is continuous for all real values of $x$, then $\mathop \sum \limits_{r = 1}^{10} \mathop {\,\,\smallint }\nolimits_0^1 \left( {r - 1 + x} \right)\,\,dx$ is equal to

$\mathop \smallint \nolimits_0^{10} \,\,f\left( x \right)\,\,dx$

$\mathop \smallint \nolimits_0^1 \,\,f\left( x \right)\,\,dx$

$10\,\,\mathop \smallint \nolimits_0^1 \,\,f\left( x \right)\,\,dx$

$9\mathop {\,\,\smallint }\nolimits_0^1 \,\,f\left( x \right)\,\,dx$

Back to DEFINITE INTEGRATION

Your Progress

0 / 791 Answered

Correct

Incorrect

Accuracy

JEE • MATHEMATICS • DEFINITE INTEGRATION

Definite integration practice · Easy

Try a question, then open “View answer” to check it. Sign in to submit answers and save progress.

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 61-90 of 791 questions

61 Easy

61 of 791

Understanding +4 | -1

\int \nolimits_{ - 10}^{10} \,\,\log \,\,\left( {\frac{{a\,\, + \,\,x}}{{a\,\, - \,\,x}}} \right)\,\,dx

is equal to

View answer and explanation

Correct option: A

\mathrm{0}

Option A (0) is correct because the integrand is an odd function and the limits of integration are symmetric about the origin.

62 Easy

62 of 791

Analyzing +4 | -1

The value of

\int_0^1 \frac{x^4(1-x)^4}{1+x^2} d x

, is

View answer and explanation

Correct option: A

\frac{22}{7}-\pi

Option A (

\frac{22}{7}-\pi

) is correct because after polynomial division and integration, the result matches this value.

63 Easy

63 of 791

Application +4 | -1

\int\limits_{0}^{\pi / 2} \frac{\cos x d x}{6-5 \sin x+\sin ^2 x}=\log K

then

K

is equal to

View answer and explanation

Correct option: B

\frac{4}{3}

Option B (4/3) is correct because the integral evaluates to log(4/3), thus K = 4/3.

64 Easy

64 of 791

Analyzing +4 | -1

The value of

\int {e^{2x}}\,(2\,\,\sin \,\,3x\,\, + \,\,3\,\,\cos \,\,3x)\,dx

View answer and explanation

Correct option: A

{e^{2x}}\,\,\sin \,\,3x\,\, + \,\,c

Option A (

{e^{2x}}\sin \\,3x\\, + \\,c

) is correct because the integral can be solved by recognizing a pattern where the cosine term cancels out after integration by parts.

65 Easy

65 of 791

Application +4 | -1

The value of

\int\limits_{0}^1 x^3(1-x)^{11} \mathrm{~d} x

is equal to

View answer and explanation

Correct option: D

\frac{1}{11}-\frac{3}{12}+\frac{3}{14}-\frac{1}{14}

Option D (

\frac{1}{11}-\frac{3}{12}+\frac{3}{13}-\frac{1}{14}

) is correct because using the property of definite integrals and expanding the polynomial simplifies the integral to the given expression.

66 Easy

66 of 791

Analyzing +4 | -1

Area bounded by

y=f^{-1}(x)

and tangent and normal drawn to it at the points with abscissae

\pi

and

2 \pi

, where

f(x)=\sin x-x

View answer and explanation

Correct option: B

\frac{\pi^2}{2}-2

Option B (

\frac{{\pi^{2}}}{{2}}-2

) is correct because the area bounded by the inverse function and the tangent and normal lines is calculated by integrating the difference between the function and the lines.

67 Easy

67 of 791

Application +4 | -1

f(x)=\left\{\begin{array}{cl}e^{\cos x} \sin x & \text { for }|x| \leq 2 \\ 2 & \text { otherwise, }\end{array}\right.

then

\int\limits_{{-2}}^3 f(x) d x=

View answer and explanation

Correct option: C

Option C is correct because the integral is split into two parts, and the first part is zero due to the integrand being an odd function, leaving only the second part to be evaluated.

68 Easy

68 of 791

Application +4 | -1

Suppose a continuous function

f:[0, \infty) \rightarrow {R}

satisfies

f(x)=2 \int\limits_{0}^x t f(t) d t+1 \text { for all } x \geq 0.

Then

f(1)

equals

View answer and explanation

Correct option: A

e

Option A (e) is correct because the function f(x) is found to be

e^{x^2}

, and thus f(1) = e.

69 Easy

69 of 791

Application +4 | -1

The value of the integral

\int\limits_{0}^{\pi / 4} \frac{\sin x+\cos x}{3+\sin 2 x} d x

, is

View answer and explanation

Correct option: C

\frac{1}{4} \log 3

Option D is correct. The integral is solved by using substitution and simplifying the expression, resulting in (1/8)log 3.

70 Easy

70 of 791

Analyzing +4 | -1

\int\limits_{0}^1 \sqrt{\frac{1-x}{1+x}} d x=

View answer and explanation

Correct option: B

\frac{\pi}{2}-1

Option B (

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

) is correct because by rationalizing the denominator and using basic integration formulas, the integral evaluates to this value.

71 Easy

71 of 791

Application +4 | -1

\int\limits_{0}^{15}\left[x^2\right] d x

is equal to

View answer and explanation

Correct option: B

2-\sqrt{2}

Option B (

2-\sqrt{2}

) is correct because the integral of the greatest integer function of x squared from 0 to 1.5 is equal to

2-\sqrt{2}

72 Easy

72 of 791

Application +4 | -1

Given

f^{\prime}(x)=f^{\prime}(2-x)

for all

x \in[0,2]

and

f(0)=1, f(2)=e^2

. Then,

\int_0^2 f(x) d x

is equal to

View answer and explanation

Correct option: D

e^2+1

The correct answer is D (

e^2+1

) because after using integration by parts and the given condition

f'(x)=f'(2-x)

, the integral evaluates to

e^2+1

73 Easy

73 of 791

Analyzing +4 | -1

\mathop {\mathrm{lim}}\limits_{n \to \infty } \,\,\left[ {{{\frac{1}{{{n^2}}}}^{}}{{\sec }^2}\,\,\frac{1}{{{n^2}}}\,\, + \,\,\frac{2}{{{n^2}}}\,\,{{\sec }^2}\,\,\frac{4}{{{n^2}}}\,\, + ... + \,\,\frac{n}{{{n^2}}}\,\,{{\sec }^2}\,\,1} \right]

equals

View answer and explanation

Correct option: A

\frac{1}{2}\,\,\tan \,\,1

Option A (

\frac{1}{2}\,\tan \,1

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

\frac{1}{2} \tan 1

74 Easy

74 of 791

Application +4 | -1

f\left( x \right) = \int \nolimits_x^{{x^2}} \left( {t - 1} \right)\,\,dt,\;\,\,1 \le \,\,\,x\,\, \le 2

, then global maximum value of

f\left( x \right)

View answer and explanation

Correct option: D

\mathrm{4}

Option D (4) is correct because the function f(x) is increasing in the given interval, so the global maximum occurs at x=2, and f(2) = 4.

75 Easy

75 of 791

Application +4 | -1

The value of

\int \limits_\pi ^{2\pi } \,\,\left[ {2\,\,\sin \,\,x} \right]dx

, where

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

represents the greatest integer function, is

View answer and explanation

Correct option: A

\frac{{ - \,5\,\pi }}{3}

Option A (

\frac{{ - \,5\pi }}{3}

) is correct because the integral of the greatest integer function of 2sin(x) from pi to 2pi is calculated by splitting the integral into intervals where the greatest integer function has a constant value.

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

76 Easy

76 of 791

Analyzing +4 | -1

\int_0^x\{t\} d t=\int_0^{\{x\}} t d t

(where

x>0, x \notin Z

and {.} represents fractional part function), then

View answer and explanation

Correct option: A

x \in(0,1)

Option A (

x \in(0,1)

) is correct because the equation holds true only when x lies in the interval (0, 1).

77 Easy

77 of 791

Application +4 | -1

The value of

\int\limits_{{-\pi / 2}}^{\pi / 2}\left(3 \sin x+\sin ^3 x\right) d x

View answer and explanation

Correct option: C

Option C (0) is correct because the integrand (3sin(x) + sin³(x)) is an odd function, and the integral is evaluated over a symmetric interval [-π/2, π/2].

78 Easy

78 of 791

Analyzing +4 | -1

I\, = \,\int_1^2 {\frac{{dx}}{{\sqrt {2{x^3} - 9{x^2} + 12x + 4} }},}

then

View answer and explanation

Correct option: C

\frac{1}{9} < {I^2} < \frac{1}{8}

Option C (

\frac{1}{9} < {I^2} < \frac{1}{8}

) is correct because it accurately bounds the square of the integral based on the function's increasing nature within the given interval.

79 Easy

79 of 791

Application +4 | -1

The value of

\lim \limits_\limits {x \rightarrow 0} \frac{\int\limits_{0}^{x^2} \sec ^2 t d t}{x \sin x}

View answer and explanation

Correct option: D

Option D (1) is correct because L'Hopital's rule is applied to evaluate the limit.

80 Easy

80 of 791

Application +4 | -1

Let

f:[-1,0] \rightarrow R

be a function differentiable within the domain and that

\int\limits_{{-1}}^0(f(x))^2 d x=10

and

f(-1)=2

. The value of the integral

\int\limits_{{-1}}^0 x f^{\prime}(x) f(x) d x

, is:

View answer and explanation

Correct option: C

-3

Option C is correct because integrating by parts and using the given information leads to the value -3.

81 Easy

81 of 791

Analyzing +4 | -1

The area bounded by the

x

-axis, the curve

y=f(x)

and the lines

x=1, x=b

is equal to

\sqrt{b^2+1}-\sqrt{2}

for all

b>1

, then

f(r)

View answer and explanation

Correct option: D

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

Option D (

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

) is correct because it is the derivative of

\sqrt{x^2+1}

, which is obtained by differentiating the given integral equation with respect to b.

82 Easy

82 of 791

Application +4 | -1

The value of

\int\limits_{0}^3|2-x| d x

View answer and explanation

Correct option: B

Option B (2) is correct because the integral is evaluated by splitting it into two parts based on the absolute value function and then integrating each part.

83 Easy

83 of 791

Application +4 | -1

m \neq n, m, n \in \mathbf{N}

then the value of

\int\limits_{0}^{2 \pi} \cos m x

\cos n x \mathrm{~d} x

View answer and explanation

Correct option: A

Option A (0) is correct because the integral of the product of cosine functions with different integer multiples over a full period is zero.

84 Easy

84 of 791

Application +4 | -1

The value of

\int\limits_{{1 / 2}}^1\left(2 x \sin \frac{1}{x}-\cos \frac{1}{x}\right) \mathrm{d} x

View answer and explanation

Correct option: D

None of these

Option D (None of these) is correct because the integral evaluates to

\sin(1) - \frac{1}{4}\sin(2)

, which is not among the given options.

85 Easy

85 of 791

Understanding +4 | -1

The value of

\int \limits_{ - 1}^1 \,x\,\,\left| {\,x\,} \right|dx

, is

View answer and explanation

Correct option: C

\mathrm{0}

Option C (0) is correct because the function x|x| is an odd function, and the integral of an odd function over a symmetric interval is zero.

86 Easy

86 of 791

Application +4 | -1

The value of

\int\limits_{1}^{16} \tan ^{-1} \sqrt{\sqrt{x}-1} \mathrm{~d} x

View answer and explanation

Correct option: D

\frac{16}{3} \pi-2 \sqrt{3}

Option D (

\frac{16}{3} \pi-2 \sqrt{3}

) is correct. The integral is solved using integration by parts and a suitable substitution to simplify the expression.

87 Easy

87 of 791

Analyzing +4 | -1

f\,\left( x \right)

is continuous for all real values of

x

, then

\sum \limits_{r = 1}^{10} \mathop {\,\,\int }\nolimits_0^1 \left( {r - 1 + x} \right)\,\,dx

is equal to

View answer and explanation

Correct option: A

\int \nolimits_0^{10} \,\,f\left( x \right)\,\,dx

Option A (

\int\nolimits_0^{10} \,\,f\left( x \right)\,\,dx

) is correct because the sum of the integrals simplifies to a single integral from 0 to 10 due to the continuity of f(x).

88 Easy

88 of 791

Application +4 | -1

I_n=\int\limits_{0}^1 x^n e^{-x} \, d x

for

n \in N

, then

I_n-n I_{n-1}=

View answer and explanation

Correct option: C

-1 / e

Option C (-1 / e) is correct because it is obtained by applying integration by parts to the integral and simplifying the resulting expression.

89 Easy

89 of 791

Application +4 | -1

The value of

\int\limits_{0}^\pi[2 \sin x] \mathrm{d} x

, where [,] represents the greatest integer function is

View answer and explanation

Correct option: D

2 \pi / 3

Option D (

2 \pi / 3

) is correct. The integral is solved by breaking it into intervals where the greatest integer function has constant values and then evaluating the resulting integrals.

90 Easy

90 of 791

Application +4 | -1

The value of

I=\int\limits_{0}^1 x\left|x-\frac{1}{2}\right| d x

View answer and explanation

Correct option: C

1 / 8

Option C (1/8) is correct because the integral is solved by splitting the interval at 1/2 due to the absolute value, and then integrating each part separately.

Revision notes and key takeaways

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

Showing 61-90 of 791 questions

Continue Your Practice

📚 Switch Set (EASY)

Chapter Sets

ALL 995 MCQs EASY 791 MCQs HARD 51 MCQs MEDIUM 596 MCQs

0 /791

0✓ 0✗

0% Accuracy