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JEE • MATHEMATICS • DEFINITE INTEGRATION

Definite integration practice · Mixed difficulty

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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 931-960 of 995 questions

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931 Easy
931 of 995
Application +4 | -1
The value of lim⁡n→∞1n∑r=02n−1n2n2+4r2\lim\limits_{n \to \infty } \frac{1}{n}\sum\limits_{r = 0}^{2n - 1} {\frac{{{n^2}}}{{{n^2} + 4{r^2}}}} is
View answer and explanation
Correct option: B
12tan⁡−1(4)\frac{1}{2}{\tan ^{ - 1}}(4)
Option B is correct because the limit of the Riemann sum can be expressed as a definite integral, which evaluates to 12tan⁡−1(4)\frac{1}{2}{\tan ^{ - 1}}(4).
932 Easy
932 of 995
Application +4 | -1
∫−1/720/7  sin⁡  (x−[x])  dx\int \nolimits_{ - 1/7}^{20/7} \,\,\sin \,\,\left( {x - \left[ x \right]} \right)\,\,dx (where [ ⋅ ]\left[ {\, \cdot \,} \right] denotes greatest integer) is
View answer and explanation
Correct option: C
3  (1−  cos⁡  1)3\,\,(1 - \,\,\cos \,\,1)
Option C (3  (1−  cos⁡  1)3\,\,(1 - \,\,\cos \,\,1)) is correct because the integral is evaluated using the property of greatest integer function and then integrating.
933 Easy
933 of 995
Application +4 | -1
If ∫extf(t)dt=sin⁡x−xcos⁡x−x22\int_e^x t f(t) d t=\sin x-x \cos x-\frac{x^2}{2} for all x∈R−{0}x \in R-\{0\}, then the value of f(π6)f\left(\frac{\pi}{6}\right) will be equal to
View answer and explanation
Correct option: C
−12-\frac{1}{2}
Option C (-1/2) is correct because differentiating the given integral equation using Leibniz's rule allows us to solve for f(x), and then evaluating f(π/6) yields -1/2.
934 Easy
934 of 995
Analyzing +4 | -1
If In=∫0π/4tan⁡n  x  dx{I_n} = \int \limits_0^{\pi /4} {\tan ^n}\,\,x\,\,dx, then lim⁡n→∞  n  (In+1  +  In−1)\lim\limits_{n \to \infty } \,\,n\,\,\left( {{I_{n + 1}}\,\, + \,\,{I_{n - 1}}} \right) equals
View answer and explanation
Correct option: A
1
Option A (1) is correct because the limit simplifies to 1 by using the recurrence relation In+1 + In-1 = 1/n.
935 Easy
935 of 995
Application +4 | -1
If the tangent to the graph function y=f(x)y = f\left( x \right) makes angles π4\frac{\pi }{4} and π3\frac{\pi }{3} with the xx-axis is at the point x=2x = 2 and x=4x = 4 respectively, the value of ∫24 f′(x)  f′′(x)    dx\int \nolimits_2^4 \,f'\left( x \right)\,\,f''\left( x \right)\,\,\;dx
View answer and explanation
Correct option: D
1\mathrm{1}
Option D (1\mathrm{1}) is correct because it is the result of evaluating the definite integral using the given tangent values.
936 Medium
936 of 995
Analyzing +4 | -1
If f(x)=lim⁡n→∞  n2(x1/n  −  x1/(n+1)f\left( x \right) = \lim\limits_{n \to \infty } \,\,{n^2}({x^{1/n}}\,\, - \,\,{x^{1/\left( {n + 1} \right)}}, x>0,   then  ∫x  f(x)  dxx > 0,\,\;\mathrm{then}\,\,\int x\;f\left( x \right)\,\,dx is equal to

View answer and explanation
Correct option: D
None of these
Option B (π4\frac{\mathrm{\pi }}{4}) is correct because it is the result of the definite integral after the trigonometric substitution and simplification.
937 Medium
937 of 995
Application +4 | -1
If I=∫01ett+1dtI=\int\limits_{0}^1 \frac{e^t}{t+1} d t, then ∫a−1ae−tt−a−1dt=\int\limits_{{a-1}}^a \frac{e^{-t}}{t-a-1} d t=
View answer and explanation
Correct option: C
(−I)e−a(-I) e^{-a}
Option C ((-I) e−ae^{-a}) is correct because after substitution and simplification, the integral transforms to (−I)e−a(-I) e^{-a}.
938 Easy
938 of 995
Analyzing +4 | -1
If
I1=∫01 2x2 dx,   I2=∫ 012x3 dx,   I3=∫12 2x2dx  and  I4=∫12 2x3 dx{I_1} = \int \limits_0^1 \,{2^{{x^2}}}\,dx,\,\;{I_2} = \mathop {\int \,}\limits_0^1 {2^{{x^3}}}\,dx,\;\,{I_3} = \int \limits_1^2 \,{2^{{x^2}}}dx\,\,\mathrm{and}\,\,{I_4} = \int \limits_1^2 \,{2^{{x^3}}}\,dx
Then
View answer and explanation
Correct option: A
I1>I2{I_1} > {I_2} and I4>I3{I_4} > {I_3}
Option C (I1>I2{I_1} > {I_2} and I3>I4{I_3} > {I_4}) is correct because for 0 < x < 1, x² > x³, and for 1 < x < 2, x³ > x².
939 Easy
939 of 995
Analyzing +4 | -1
For I(x)=∫sec⁡2x−2022sin⁡2022xdxI(x)=\int \frac{\sec ^2 x-2022}{\sin ^{2022} x} d x, if I(π4)=21011I\left(\frac{\pi}{4}\right)=2^{1011}, then
View answer and explanation
Correct option: A
31010I(π3)−I(π6)=03^{1010} I\left(\frac{\pi}{3}\right)-I\left(\frac{\pi}{6}\right)=0
Option A (31010I(π3)−I(π6)=03^{1010} I(\frac{\pi}{3})-I(\frac{\pi}{6})=0) is correct because after integrating and substituting the given values, the equation holds true.
940 Easy
940 of 995
Application +4 | -1
The value of the integral ∫03  dxx+1 +5x+1\int \limits_0^3 \,\,\frac{{dx}}{{\sqrt {x + 1} \, + \sqrt {5x + 1} }}
View answer and explanation
Correct option: D
None of these
The correct answer is D (None of these) because the integral evaluates to a different expression than the provided options.
941 Easy
941 of 995
Application +4 | -1
Let f(x)=∫1x2−t2dtf(x)=\int\limits_{1}^x \sqrt{2-t^2} d t. Then the real roots of the equation x2−f′(x)=0x^2-f^{\prime}(x)=0 are
View answer and explanation
Correct option: A
±1\pm 1
Option A (±1\pm 1) is correct because these values satisfy the equation x2=2−x2x^2 = \sqrt{2-x^2} derived from the given conditions.
942 Medium
942 of 995
Application +4 | -1
The value of ∫0π∣cos⁡x∣dx\int\limits_{0}^\pi|\cos x| d x, is
View answer and explanation
Correct option: B
2
Option B (2) is correct because the integral of the absolute value of cos(x) from 0 to pi is equal to 2.
943 Hard
943 of 995
Application +4 | -1
The value of the integral ∫1/ee∣log⁡  x∣dx\int \limits_{1/e}^e \left| {\log \,\,x} \right|dx, is
View answer and explanation
Correct option: A
2  (e−1e)2\,\,\left( {\frac{{e - 1}}{e}} \right)
Option A (2  (e−1e)2\,\,\left( {\frac{{e - 1}}{e}} \right)) is correct because the integral of the absolute value of log x from 1/e to e evaluates to this expression.
944 Easy
944 of 995
Application +4 | -1
∫−ππ(cos⁡px−sin⁡qx)2 dx\int\limits_{{-\pi}}^\pi(\cos p x-\sin q x)^2 \mathrm{~d} x, where pp and qq are integers is equal to
View answer and explanation
Correct option: D
2π2 \pi
Option D (8) is correct because by solving the given equation for [x/8], we find the interval for x and then calculate b-a.
945 Easy
945 of 995
Application +4 | -1
The value of ∫−π/4π/4(x∣x∣+sin⁡3x+xtan⁡2x+1)dx is \int_{-\pi / 4}^{\pi / 4}\left(x|x|+\sin ^3 x+x \tan ^2 x+1\right) d x \text { is }
View answer and explanation
Correct option: D
π/2\pi / 2
Option D (π/2\pi / 2) is correct because the integral of the odd functions over the symmetric interval is zero, and the integral of 1 from −π/4-\pi/4 to π/4\pi/4 is π/2\pi/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
946 Easy
946 of 995
Application +4 | -1
If F  (x)=∫x2x3  log⁡  t  dt  (x>0)F\,\,\left( x \right) = \int \limits_{{x^2}}^{{x^3}} \,\,\log \,\,t\,\,dt\,\,(x > 0), then F′(x)F'\left( x \right) equals
View answer and explanation
Correct option: A
(9 x2  −  4 x)  log⁡  x\left( {9\,{x^2}\,\, - \,\,4\,x} \right)\,\,\log \,\,x
Option A is correct because the derivative of the integral is found using Leibniz's rule, resulting in (9x² - 4x)log x.
947 Easy
947 of 995
Analyzing +4 | -1
The value of the integral ∫02π  sin⁡  2θa−b  cos⁡  θ  dθ\int \limits_0^{2\pi } \,\,\frac{{\sin \,\,2\theta }}{{a - b\,\,\cos \,\,\theta }}\,\,d\theta when a   >   b   >   0a\,\,\, > \,\,\,b\,\,\, > \,\,\,0, is
View answer and explanation
Correct option: D
1\mathrm{1}
Option A is correct because the integral evaluates to 0 due to the symmetry of the integrand over the interval [0, 2π].
948 Easy
948 of 995
Application +4 | -1
The solution of the equation ∫2xdxxx2−1=π12\int\limits_ {\sqrt{2}}^x \frac{d x}{x \sqrt{x^2-1}}=\frac{\pi}{12} is given by
View answer and explanation
Correct option: B
2
The correct answer is B (2) because it satisfies the given equation after evaluating the integral and applying the inverse secant function.
949 Medium
949 of 995
Application +4 | -1
If [x]\left[ x \right] is the greatest integer ≤x\le x then π2∫02(sin⁡πx2)(x−∣x∣)[x]dx{\pi ^2}\int\limits_0^2 {\left( {\sin \frac{{\pi x}}{2}} \right)(x - \left| x \right|} {)^{\left[ x \right]}}dx is equal to
View answer and explanation
Correct option: B
4(π−1)4\left( {\pi - 1} \right)
Option B (4(π−1)4(\pi - 1)) is correct because the integral is evaluated by splitting the integration range and using the greatest integer function.
950 Easy
950 of 995
Application +4 | -1
The value of lim⁡x→0∫0x2sin⁡tdtx3\lim \limits_{{x \rightarrow 0}} \frac{\int\limits_{0}^{x^2} \sin \sqrt{t} d t}{x^3}
View answer and explanation
Correct option: B
23\frac{2}{3}
Option B (23\frac{2}{3}) is correct because it is the result of applying L'Hopital's rule and Leibniz's rule to evaluate the limit.
951 Hard
951 of 995
Analyzing +4 | -1
If f  (x)f\,\,\left( x \right) is a function satisfying f(1x)  +  x2 f(x)=0f\left( {\frac{1}{x}} \right)\,\, + \,\,{x^2}\,f\left( x \right) = 0 for all non-zero xx, then ∫sin⁡  θcosecθ  f(x)  dx\int \nolimits_{\sin \,\,\mathrm{\theta }}^{\mathrm{cosec \theta }} \,\,f\left( x \right)\,\,dx is equal to
View answer and explanation
Correct option: A
0\mathrm{0}
Option A (0) is correct because using the substitution x = 1/t and the given functional equation, the integral becomes its negative, implying it equals zero.
952 Easy
952 of 995
Application +4 | -1
The value of the integral ∫02a  f(x)f(x)  +  f(2a−x)  dx\int \limits_0^{2a} \,\,\frac{{f\left( x \right)}}{{f\left( x \right)\,\, + \,\,f\left( {2a - x} \right)}}\,\,dx is equal to
View answer and explanation
Correct option: C
aa
Option C (aa) is correct because the integral simplifies to 'a' after applying the property of definite integrals.
953 Easy
953 of 995
Analyzing +4 | -1
The integral ∫0π1+4sin⁡2x2−4sin⁡x2 dx\int\limits_{0}^\pi \sqrt{1+4 \sin ^2 \frac{x}{2}-4 \sin \frac{x}{2}} \,d x equals
View answer and explanation
Correct option: B
43−4−π34 \sqrt{3}-4-\frac{\pi}{3}
Option B (43−4−π34 \sqrt{3}-4-\frac{\pi}{3}) is correct because it is obtained by simplifying the integral using trigonometric identities and evaluating the resulting integral.
954 Easy
954 of 995
Application +4 | -1
∫0π2/4(2cosec⁡x+xsec⁡x)dx=\int\limits_{0}^{\pi^2 / 4}\left(\frac{2}{\operatorname{cosec} \sqrt{x}}+\frac{\sqrt{x}}{\sec \sqrt{x}}\right) d x=
View answer and explanation
Correct option: B
π22\frac{\pi^2}{2}
Option B, π22\frac{\pi^2}{2}, is correct because it results from evaluating the integral using the given function and its derivative.
955 Medium
955 of 995
Application +4 | -1
The value of the integral ∫01xα−1ln⁡xdx\int\limits_{0}^1 \frac{x^\alpha-1}{\ln x} d x, is
View answer and explanation
Correct option: D
None of these
Option D (None of these) is correct because the correct result of the integral is ln(α\alpha+1), which is not among the options.
956 Medium
956 of 995
Analyzing +4 | -1
If ∫01  cot⁡−1  (1  −  x  +  x2)  dx=k ∫01tan⁡−1  x dx\int \limits_0^1 \,\,{\cot ^{ - 1}}\,\,\left( {1\,\, - \,\,x\,\, + \,\,{x^2}} \right)\,\,dx = k\,\int \limits_0^1 {\tan ^{ - 1}}\,\,x\,dx, then k=k =
View answer and explanation
Correct option: B
2\mathrm{2}
Option B is correct because after applying properties of inverse trigonometric functions and simplifying, k=2k = 2.
957 Easy
957 of 995
Application +4 | -1
The mean value of the function f(x)=2ex+1f(x)=\frac{2}{e^x+1} on the interval [0,2][0,2] is
View answer and explanation
Correct option: C
2+log⁡2e2+12+\log \frac{2}{e^2+1}
Option C (2+log(2/(e²+1))) is correct because it is the result of integrating the function over the given interval and dividing by the length of the interval.
958 Medium
958 of 995
Application +4 | -1
∫−π/2π/2  sin⁡  ∣ x ∣  dx\int \nolimits_{ - \pi /2}^{\pi /2} \,\,\sin \,\,\left| {\,x\,} \right|\,\,dxis equal to
View answer and explanation
Correct option: C
2\mathrm{2}
Option C (2) is correct because the integral of sin|x| from -π\pi/2 to π\pi/2 is twice the integral of sin(x) from 0 to π\pi/2.
959 Easy
959 of 995
Understanding +4 | -1
The value of the integral ∫0π/2  (sin⁡100  x−  cos⁡100  x)  dx\int \nolimits_0^{\pi /2} \,\,({\sin ^{100}}\,\,x - \,\,{\cos ^{100}}\,\,x)\,\,dx is
View answer and explanation
Correct option: D
0
Option D (0) is correct because the integral of sin100(x) from 0 to pi/2 is equal to the integral of cos100(x) from 0 to pi/2, resulting in a difference of 0.
960 Easy
960 of 995
Application +4 | -1
∫0π/4  sin⁡  x  +  cos⁡  x3  +  sin⁡  2x  dx\int \limits_0^{\pi /4} \,\,\frac{{\sin \,\,x\,\, + \,\,\cos \,\,x}}{{3\,\, + \,\,\sin \,\,2x}}\,\,dx is equal to
View answer and explanation
Correct option: B
14  log⁡  3\frac{1}{4}\,\,\log \,\,3
Option B (1/4 log 3) is correct because after substitution and integration, the definite integral evaluates to (1/4)log 3.
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