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JEE • MATHEMATICS • DEFINITE INTEGRATION

Definite integration practice · Medium

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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 596 questions
1 Medium
1 of 596
Application +4 | -1
If f(x)f\left( x \right) is differentiable and ∫0t2  x   f(x)  dx=25  t5\int \nolimits_0^{{t^2}} \,\,x\;\,f\left( x \right)\,\,dx = \frac{2}{5}\,\,{t^5}, then f(425)f\left( {\frac{4}{{25}}} \right) equals
View answer and explanation
Correct option: A
25\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.
2 Medium
2 of 596
Application +4 | -1
The value of ∫011[x]3.dx\int\limits_{{0}}^{11}[x]^{3} . \mathrm{dx}, where [∗][*] denotes the greatest integer function, is
View answer and explanation
Correct option: D
3025
Option D (3025) is correct because the integral of the greatest integer function cubed from 0 to 11 is the sum of cubes from 0 to 10, which equals 3025.
3 Medium
3 of 596
Analyzing +4 | -1
The integral
∫24log⁡x2log⁡x2+log⁡(36−12x+x2)dx\int\limits_{2}^4 \frac{\log x^2}{\log x^2+\log \left(36-12 x+x^2\right)} d x is equal to:
View answer and explanation
Correct option: A
1
Option A (1) is correct because by using the property of definite integrals and adding the two forms of the integral, the result simplifies to 1.
4 Medium
4 of 596
Analyzing +4 | -1
If I  (m,  n)=∫01  tm  (1+t)n  dtI\,\,\left( {m,\,\,n} \right) = \int \nolimits_0^1 \,\,{t^m}\,\,{\left( {1 + t} \right)^n}\,\,dt,then the expression for I(m,  n)I\left( {m,\,\,n} \right) in terms of I  (m+1,  n+1)I\,\,\left( {m + 1,\,\,n + 1} \right) is
View answer and explanation
Correct option: A
2nm+1  −   nm+1  (m+1,n−1)\frac{{{2^n}}}{{m + 1}}\,\, - \,\,\,\frac{n}{{m + 1}}\,\,\left( {m + 1,n - 1} \right)
Option A (2nm+1 −   nm+1   (m+1,n−1)\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 596
Analyzing +4 | -1
If I=∫34  1log⁡  x3  dxI = \int \limits_3^4 \,\,\frac{1}{{\sqrt[3]{{\log \,\,x}}}}\,\,dx, then
View answer and explanation
Correct option: A
0.92<I<10.92 < I < 1
Option A (0.92<I<10.92 < I < 1) is correct because the integral is bounded by comparing the integrand with simpler functions.
6 Medium
6 of 596
Analyzing +4 | -1
If a≠ba \neq b and af(x)+bf(1x)=1x−5a f(x)+b f\left(\frac{1}{x}\right)=\frac{1}{x}-5 for all x≠0x \neq 0, then ∫12f(x)dx=1a2−b2[a(log⁡2−α)+β(b2)]\int\limits_{1}^2 f(x) d x=\frac{1}{a^2-b^2}\left[a(\log 2-\alpha)+\beta\left(\frac{b}{2}\right)\right] where β−α\beta-\alpha is equal to
View answer and explanation
Correct option: D
2
Option D (2) is correct because after solving the system of equations and integrating, the expression simplifies to the given form, and comparing coefficients reveals that β−α=2\beta - \alpha = 2.
7 Medium
7 of 596
Analyzing +4 | -1
lim⁡n→∞  {(1+  1n)  (1+  2n)  (1+  3n)  …  (1+  nn)}1/n\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/e4/e
Option B (4/e4/e) is correct because taking the logarithm and converting the sum to an integral allows us to evaluate the limit.
8 Medium
8 of 596
Application +4 | -1
The value of ∫0100ax−[x]dx\int\limits_{0}^{100} a^{x-[x]} d x is
View answer and explanation
Correct option: A
100(a−1)log⁡a\frac{100(a-1)}{\log a}
Option A (100(a−1)log⁡a\frac{{100(a-1)}}{{\log a}}) is correct because the integral simplifies to a sum of integrals over unit intervals, each of which evaluates to a−1log⁡a\frac{{a-1}}{{\log a}}.
9 Medium
9 of 596
Application +4 | -1
∫−11log⁡(1+x1−x)dx=\int\limits_{{-1}}^1 \log \left(\frac{1+x}{1-x}\right) d x=
View answer and explanation
Correct option: C
0
Option C (0) is correct because the integrand is an odd function, and the integral is evaluated over a symmetric interval.
10 Medium
10 of 596
Application +4 | -1
The value of ∫−12∣[x]−{x}∣dx\int_{-1}^2|[x]-\{x\}| d x, where [x][x] is the greatest integer less than or equal to xx and {x}\{x\} is the fractional part of xx, is
View answer and explanation
Correct option: B
5/25 / 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.
11 Medium
11 of 596
Understanding +4 | -1
If f(t)f\left( t \right) is a continuous function defined on [a,   b]\left[ {a,\;\,b} \right] such that f(t)f\left( t \right) is an odd function, then the function ϕ  (x)=∫ax  f(t)dt\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.
12 Medium
12 of 596
Application +4 | -1
Evaluate ∫01/2xsin⁡−1x1−x2dx.\int\limits_{0}^{1 / 2} \frac{x \sin ^{-1} x}{\sqrt{1-x^2}} d x.
View answer and explanation
Correct option: A
6−π312\frac{6-\pi \sqrt{3}}{12}
Option A (6−π312\frac{{6-\pi \sqrt{{3}}}}{{12}}) is correct because the integral is solved using trigonometric substitution and integration by parts.
13 Medium
13 of 596
Application +4 | -1
Let f(x)=∫xx2(t−1)dtf(x)=\int\limits_{x}^{x^2}(t-1) d t. Then the value of ∣f′(ω)∣\left|f^{\prime}(\omega)\right| where ω\omega is a complex cube root of unity is
View answer and explanation
Correct option: C
333 \sqrt{3}
Option C (333 \sqrt{3}) is correct because it is the absolute value of the derivative of the function evaluated at a complex cube root of unity.
14 Medium
14 of 596
Application +4 | -1
The value of the integral ∫1e (log⁡  x)3dx\int \limits_1^e \,{\left( {\log \,\,x} \right)^3}dx is
View answer and explanation
Correct option: B
6  −  2e6\,\, - \,\,2e
Option C (2e−62e\\ - \\ 6) is correct because the integral is solved using integration by parts repeatedly.
15 Medium
15 of 596
Application +4 | -1
∫1e3dxx1+log⁡ex=\int\limits_{1}^{e^3} \frac{d x}{x \sqrt{1+\log _e x}}=
View answer and explanation
Correct option: A
2
Option A is correct because by substituting log⁡ex=t\log_e x = t, the integral simplifies to ∫03dt1+t\int_0^3 \frac{dt}{\sqrt{1+t}}, which evaluates to 2.

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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 Medium
16 of 596
Application +4 | -1
The value of the integral ∫0π/4  sin⁡  θ  +  cos⁡  θ9  +  16  sin⁡  2θ  dθ\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
120  log⁡  3\frac{1}{{20}}\,\,\log \,\,3
Option C (120 log⁡,3\frac{1}{{20}}\\\,\log \\,3) is correct because it is the value obtained after evaluating the definite integral using substitution.
17 Medium
17 of 596
Application +4 | -1
If I10=∫0π/2x10sin⁡xdxI_{10}=\int_0^{\pi / 2} x^{10} \sin x d x, then I10+90I8I_{10}+90 I_8 is
View answer and explanation
Correct option: B
10(π/2)910(\pi / 2)^9
Option C is correct because the reduction formula for the integral of x^n sin(x) involves integration by parts, leading to the given relationship between I_10 and I_8.
18 Medium
18 of 596
Analyzing +4 | -1
If I=∫12dx2x3−9x2+12x+4\mathrm{I}=\int_1^2 \frac{\mathrm{dx}}{\sqrt{2 \mathrm{x}^3-9 \mathrm{x}^2+12 \mathrm{x}+4}}, then
View answer and explanation
Correct option: B
19<I2<18\frac{1}{9}<\mathrm{I}^2<\frac{1}{8}
Option B (19<I2<18\frac{1}{9}<\mathrm{I}^2<\frac{1}{8}) is correct because it is found by determining the bounds of the integral based on the function's increasing behavior.
19 Medium
19 of 596
Application +4 | -1
The area of the region bounded by y=x2y=x^2 and y=−x2+2y=-x^2+2 is
View answer and explanation
Correct option: C
83\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.
20 Medium
20 of 596
Analyzing +4 | -1
The value of ∫012x3−3x2−x+13 dx\int\limits_{0}^1 \sqrt[3]{2 x^3-3 x^2-x+1} \, d x is equal to
View answer and explanation
Correct option: C
0
Option A (1) is correct because the equation holds only for a = 0, resulting in k = 1.
21 Medium
21 of 596
Analyzing +4 | -1
The value of ∫−π/2π/2dx[x]+[sin⁡x]+4\int\limits_{{-\pi / 2}}^{\pi / 2} \frac{d x}{[x]+[\sin x]+4}, where [t][t] denotes the greatest integer less than or equal to tt, is:
View answer and explanation
Correct option: C
320(4π−3)\frac{3}{20}(4 \pi-3)
Option C (320(4π−3)\frac{3}{20}(4 \pi-3)) is correct because the integral is solved by breaking it into intervals based on the greatest integer function.
22 Medium
22 of 596
Application +4 | -1
The value of kk for which the area of the figure bounded by the curve y=8  x2−x5,y = 8\;{x^2} - {x^5}, the straight line x=1x = 1 and x=kx = k and the xx-axis is equal to 16/3
View answer and explanation
Correct option: B
8−173\sqrt[3]{{8 - \sqrt {17} }}
The correct answer is B (8−173\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.
23 Medium
23 of 596
Analyzing +4 | -1
The value of
∫01(∏r=1n(x+r))(∑k=1n1x+k)dx\int_0^1\left(\prod_{r=1}^n(x+r)\right)\left(\sum\limits_{{k=1}}^n \frac{1}{x+k}\right) d x equals to
View answer and explanation
Correct option: D
n⋅nn \cdot n !
Option D (n⋅n!n \cdot n !) is correct because the integral simplifies to the difference of the product of consecutive integers from 1 to n+1 and 1 to n, which equals n⋅n!n \cdot n !.
24 Medium
24 of 596
Application +4 | -1
Let f(x)f\left( x \right) be a function satisfying f′  (x)=f  (x)f'\,\,\left( x \right) = f\,\,\left( x \right) with f(0)=1f\left( 0 \right) = 1 and g(x)g\left( x \right) be the function satisfying f(x)  +  g(x)=x2f\left( x \right)\,\, + \,\,g\left( x \right) = {x^2}. The, value of the integral∫01 f(x)  g  (x)dx\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−12e2−32e - \frac{1}{2}{e^2} - \frac{3}{2}.
25 Medium
25 of 596
Analyzing +4 | -1
If I1=∫01e−xcos⁡2xdx;I2=∫01e−x2cos⁡2xdxI_1=\int\limits_{0}^1 e^{-x} \cos ^2 x d x ; I_2=\int\limits_{0}^1 e^{-x^2} \cos ^2 x d x and I3=∫01e−x3I_3=\int\limits_{0}^1 e^{-x^3} dx;d x ; then
View answer and explanation
Correct option: D
I3>I2>I1I_3 > I_2 > I_1
Option D (I3>I2>I1I_3 > I_2 > I_1) is correct because the inequality e−x3>e−x2>e−xe^{-x^3} > e^{-x^2} > e^{-x} holds for x∈(0,1)x \in (0,1).
26 Medium
26 of 596
Application +4 | -1
The value of the integral ∫1/22tan⁡−1xxdx\int\limits_{{1 / 2}}^2 \frac{\tan ^{-1} x}{x} d x is equal to
View answer and explanation
Correct option: C
π2log⁡e2\frac{\pi}{2} \log _e 2
Option C (π2log⁡e2\frac{\pi}{2} \log _e 2) is correct because by using the property of definite integrals and substituting xx with 1x\frac{1}{x}, we can simplify the integral and solve it.
27 Medium
27 of 596
Analyzing +4 | -1
∫−11cos⁡xlog⁡e(1−x1+x)dx=\int\limits_{{-1}}^1 \sqrt{\cos x} \log _e\left(\frac{1-x}{1+x}\right) d x=
View answer and explanation
Correct option: A
0
Option A, 0, is correct because the integrand is an odd function, and the integral is evaluated over a symmetric interval [-1, 1].
28 Medium
28 of 596
Application +4 | -1
Assume that ff is continuous everywhere, then 1c∫xbcf(xc)dx=\frac{1}{c} \int\limits_{x}^{b c} f\left(\frac{x}{c}\right) d x=
View answer and explanation
Correct option: C
∫abf(x)dx\int\limits_{a}^b f(x) d x
Option C (∫abf(x)dx\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.
29 Medium
29 of 596
Application +4 | -1
The set of values of ' aa ' which satisfy the equation ∫02(t−log⁡2a)dt=log⁡2(4a2)\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∈R+a \in R^{+}
Option B is correct because the equation simplifies to an identity, implying that the solution set is all positive real numbers.
30 Medium
30 of 596
Application +4 | -1
The area of the figure bounded by y=sin⁡x,y=cos⁡xy = \sin x,y = \cos x in the first quadrant, is
View answer and explanation
Correct option: A
2(2−1)2\left( {\sqrt 2 - 1} \right)
Option A (2(2−1)2\left( {\sqrt 2 - 1} \right)) is correct because it represents the sum of the definite integrals of (cos⁡x−sin⁡x)(\cos x - \sin x) from 0 to π4\frac{\pi}{4} and (sin⁡x−cos⁡x)(\sin x - \cos x) from π4\frac{\pi}{4} to π2\frac{\pi}{2}.
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