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JEE • MATHEMATICS • DEFINITE INTEGRATION

Definite integration practice · Easy

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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.

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Showing 31-60 of 791 questions
31 Easy
31 of 791
Application +4 | -1
The difference between the greatest and least values of the function ϕ  (x)=∫0x(t+1)  dt\phi \,\,\left( x \right) = \int \limits_0^x \left( {t + 1} \right)\,\,dt on [2,3]\left[ {\mathrm{2, 3}} \right], is
View answer and explanation
Correct option: C
7 / 2\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.
32 Easy
32 of 791
Analyzing +4 | -1
For f(x)=x4+∣x∣f(x)=x^4+|x|, let I1=∫0πf(cos⁡x)dxI_1=\int_0^\pi f(\cos x) d x and I2=∫0π/2f(sin⁡x)dxI_2=\int_0^{\pi / 2} f(\sin x) d x, then I1I2\frac{I_1}{I_2} has the value equal to
View answer and explanation
Correct option: C
2
Option C (2) is correct because the integral I1I_1 can be simplified using the property of even functions and then related to I2I_2.
33 Easy
33 of 791
Application +4 | -1
∫0π/2sin⁡xsin⁡x+cos⁡xdx\int\limits_{0}^{\pi / 2} \frac{\sin x}{\sin x+\cos x} d x equals
View answer and explanation
Correct option: C
π4\frac{\pi}{4}
Option C (π4\frac{\pi}{4}) is correct because using the property ∫0af(x)dx=∫0af(a−x)dx\int_0^a f(x) dx = \int_0^a f(a-x) dx simplifies the integral to π4\frac{\pi}{4}.
34 Easy
34 of 791
Application +4 | -1
If f(x)=∫0xsin⁡8t dtf(x)=\int\limits_{0}^x \sin ^8 t \mathrm{~d} t, then f(x+π)f(x+\pi) equals
View answer and explanation
Correct option: C
f(x)+f(π)f(x)+f(\pi)
Option C (f(x)+f(π)f(x)+f(\pi)) is correct because f(x+π)f(x+\pi) can be split into two integrals, one from 0 to π\pi and the other from π\pi to x+πx+\pi, which simplifies to f(x)+f(π)f(x)+f(\pi).
35 Easy
35 of 791
Application +4 | -1
Let β(m,n)=∫01xm−1(1−x)n−1dx,m,n>0.\beta(m, n)=\int\limits_{0}^1 x^{m-1}(1-x)^{n-1} d x, m, n>0. If ∫01(1−x10)20dx=a×β(b,c),\int\limits_{0}^1\left(1-{x}^{10}\right)^{20} {dx}={a} \times \beta({b}, {c}), then 100(a+b+c)100({a}+{b}+{c}) equals
View answer and explanation
Correct option: D
2120
Option D (2120) is correct because the integral is evaluated using substitution and the properties of the Beta function.
36 Easy
36 of 791
Application +4 | -1
The area enclosed by y=x3y=x^3, its normal at (1,1)(1,1) and xx axis is equal to
View answer and explanation
Correct option: A
74\frac{7}{4}
Option A is correct because it represents the area calculated by integrating the difference between the normal line and the curve y=x3 from 0 to 1.
37 Easy
37 of 791
Application +4 | -1
∫0π  θ  sin⁡  θ1  +  cos⁡2  θ  dθ\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
π24\frac{{{\pi ^2}}}{4}
Option A (π24\frac{{{\pi ^2}}}{4}) is correct because the integral is evaluated using properties of definite integrals and trigonometric substitution.
38 Easy
38 of 791
Analyzing +4 | -1
If 2∫01tan⁡−1xdx=∫01cot⁡−1(1−x+x2)dx2 \int\limits_{0}^1 \tan ^{-1} x d x=\int\limits_{0}^1 \cot ^{-1}\left(1-x+x^2\right) d x, then ∫01tan⁡−1(1−x+x2)dx\int\limits_{0}^1 \tan ^{-1}\left(1-x+x^2\right) d x is equal to:
View answer and explanation
Correct option: B
log⁡2\log 2
Option B (log 2) is correct because after simplifying the integral equation and using integration by parts, the result is log 2.
39 Easy
39 of 791
Understanding +4 | -1
The value of ∫−ππ  (1−x2)sin⁡  x  cos⁡2  x   dx  \int \nolimits_{ - \pi }^\pi \,\,\left( {1 - {x^2}} \right)\sin \,\,x\,\,{\cos ^2}\,\,x\;\,dx\; is
View answer and explanation
Correct option: A
0\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 −π-\pi to π\pi is zero.
40 Easy
40 of 791
Application +4 | -1
∫02[x2]dx\int\limits_{0}^2\left[x^2\right] d x is
View answer and explanation
Correct option: D
−2−3+5-\sqrt{2}-\sqrt{3}+5
Option B (e−1e-1) is correct because the limit of the sum is converted to a definite integral and evaluated.
41 Easy
41 of 791
Application +4 | -1
The value of the integral ∫αβ  1(x−α)  (β−x)  dx\int \limits_\alpha ^\beta \,\,\frac{1}{{\sqrt {\left( {x - \alpha } \right)\,\,\left( {\beta - x} \right)} }}\,\,dx for β   >  α\beta \,\,\, > \,\,\alpha, is
View answer and explanation
Correct option: D
π\pi
Option D (π\pi) is correct because the integral evaluates to π\pi after completing the square and using trigonometric substitution.
42 Easy
42 of 791
Application +4 | -1
The value of the integral ∫−π/3π/3xsin⁡xcos⁡2x dx\int\limits_{{-\pi / 3}}^{\pi / 3} \frac{x \sin x}{\cos ^2 x} \mathrm{~d} x is
View answer and explanation
Correct option: B
2(2π3−log⁡(2+3))2\left(\frac{2 \pi}{3}-\log (2+\sqrt{3})\right)
The correct answer is 22a2 \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.
43 Easy
43 of 791
Analyzing +4 | -1
The integral ∫012ln⁡(1+2x)1+4x2dx\int\limits_{0}^{\frac{1}{2}} \frac{\ln (1+2 \mathrm{x})}{1+4 \mathrm{x}^2} \mathrm{dx}, equals:
View answer and explanation
Correct option: C
π16ln⁡2\frac{\pi}{16} \ln 2
Option C (π16ln⁡2\frac{\pi}{16} \ln 2) is correct because it is the result of evaluating the integral using substitution and properties of definite integrals.
44 Easy
44 of 791
Analyzing +4 | -1
lim⁡n→∞(nn2+12+nn2+22+nn2+32+...+15n)\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(2){\tan ^{ - 1}}\left( 2 \right)
Option B (tan⁡−1(2){\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){\tan ^{ - 1}}(2).
45 Easy
45 of 791
Application +4 | -1
The area enclosed by the curve x225+y216=1\frac{{{x^2}}}{{25}} + \frac{{{y^2}}}{{16}} = 1 is
View answer and explanation
Correct option: B
20π20\pi sq unit
Option B, 20π20\pi sq unit, is the correct area of the ellipse because the area of an ellipse is given by πab\pi ab, where a and b are the semi-major and semi-minor axes, respectively.

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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
46 Easy
46 of 791
Application +4 | -1
For 0<α<π0 < \alpha < \pi. The value of the definite integral ∫01 dx/(x2+2xcos⁡α+1)\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
α/(2sin⁡α)\alpha /(2 \sin \alpha)
Option A (α/(2sin⁡α)\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.
47 Easy
47 of 791
Application +4 | -1
∫815  dx(x−3)  x+1\int \nolimits_8^{15} \,\,\frac{{dx}}{{\left( {x - 3} \right)\,\,\sqrt {x + 1} }} is equal to
View answer and explanation
Correct option: A
12  log⁡  53\frac{1}{2}\,\,\log \,\,\frac{5}{3}
Option A (12 log⁡ 53\frac{1}{2}\,\log \,\frac{5}{3}) is correct because after substituting x+1=t2x + 1 = t^2 and evaluating the definite integral, the result is 12log⁡53\frac{1}{2} \log \frac{5}{3}.
48 Easy
48 of 791
Application +4 | -1
If ∫0π  x  f  (sin⁡  x)  dx=A  ∫0π/2  f  (sin⁡  x)  dx,then  A\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
π\pi
Option B (π\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 π/2\pi/2, and A is found to be π\pi.
49 Easy
49 of 791
Application +4 | -1
∫−π/2π/2sin⁡2  x  cos⁡2  x  (sin⁡  x  +  cos⁡  x)  dx=\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
4/154/15
Option B (4/154/15) is correct. The integral is evaluated by splitting it into odd and even functions and using trigonometric identities.
50 Easy
50 of 791
Analyzing +4 | -1
If In=∫−ππsin⁡nx(1+πx)sin⁡xdx,   n=0,1,2,…I_n=\int\limits_{{-\pi}}^\pi \frac{\sin n x}{\left(1+\pi^x\right) \sin x} d x, \,\,\, n=0,1,2, \ldots then
View answer and explanation
Correct option: A
In=In+2I_n=I_{n+2}
Option A (In=In+2I_n=I_{n+2}) is correct because by using the property of definite integrals and trigonometric identities, it can be shown that In+2−In=0I_{n+2} - I_n = 0.
51 Easy
51 of 791
Application +4 | -1
∫0π/4sin⁡  (x  −  [x])  dx\int \limits_0^{\pi /4} \sin \,\,\left( {x\,\, - \,\,\left[ x \right]} \right)\,\,dx is equal to
View answer and explanation
Correct option: B
1−121 - \frac{1}{{\sqrt 2 }}
Option B (1 - 1/√2) is correct because the integral simplifies to the integral of sin(x) from 0 to π/4, which evaluates to 1 - 1/√2.
52 Easy
52 of 791
Analyzing +4 | -1
If [x][x] stands for the greatest integer function, the value of ∫410[x2][x2−28x+196]+[x2]dx\int\limits_{4}^{10} \frac{\left[x^2\right]}{\left[x^2-28 x+196\right]+\left[x^2\right]} d x, is
View answer and explanation
Correct option: C
3
Option C (3) is correct because the integral is of the form ∫abf(x)f(x)+f(a+b−x)dx\int_a^b \frac{f(x)}{{f(x) + f(a+b-x)}} dx, which equals b−a2\frac{{b-a}}{{2}}.
53 Easy
53 of 791
Analyzing +4 | -1
If f(x)=ex1+ex,I1=∫f(−a)f(a)xg{x(1−x)}dxf(x)=\frac{e^x}{1+e^x}, I_1=\int\limits_{{f(-a)}}^{f(a)} x g\{x(1-x)\} d x and I2=∫f(−a)f(a)g{x(1−x)}dxI_2=\int\limits_{{f(-a)}}^{f(a)} g\{x(1-x)\} d x, where gg is not an identity function. Then the value of I2/I1I_2 / I_1, is
View answer and explanation
Correct option: C
2
Option C (2) is correct because by using the property ∫abf(x)dx=∫abf(a+b−x)dx\int_a^b f(x) dx = \int_a^b f(a+b-x) dx, we find that I2=2I1I_2 = 2I_1, thus I2/I1=2I_2/I_1 = 2.
54 Easy
54 of 791
Application +4 | -1
∫0π    xdxa2  cos⁡2  x  +  b2  sin⁡2  x  is equal to\int \nolimits_0^\pi \,\,\,\,\frac{{x\mathrm{ }dx}}{{{a^2}\,\,{{\cos }^2}\,\,x\,\, + \,\,{b^2}\,\,{{\sin }^2}\,\,x}}\,\,\mathrm{is }\,\mathrm{equal }\,\mathrm{to}
View answer and explanation
Correct option: C
π22 ab\frac{{{\pi ^2}}}{{2\,ab}}
Option C (π22 ab\frac{{{\pi ^2}}}{{2\,ab}}) is correct because the integral is solved using properties of definite integrals and trigonometric substitutions.
55 Easy
55 of 791
Application +4 | -1
∫0πcos⁡3xdx=\int\limits_{0}^\pi \cos ^3 x d x=
View answer and explanation
Correct option: B
0
Option B (0) is correct because the integral of cos³(x) from 0 to π is zero due to symmetry.
56 Easy
56 of 791
Application +4 | -1
The area of the figure bounded by the curve y=2x−x2y=2 x-x^2 and the straight line y=−xy=-x is
View answer and explanation
Correct option: A
92\frac{9}{2}
Option A (92\frac{9}{2}) is correct because it represents the area between the parabola and the line, found by integrating the difference of the functions between their intersection points.
57 Easy
57 of 791
Application +4 | -1
lim⁡n→∞[11+n+12+n+13+n+…+12n]\lim \limits_{{n \rightarrow \infty}}\left[\frac{1}{1+n}+\frac{1}{2+n}+\frac{1}{3+n}+\ldots+\frac{1}{2 n}\right] is equal to
View answer and explanation
Correct option: C
log⁡e2\log _{e} 2
Option C (log⁡e2\log _{e} 2) is correct because the limit can be expressed as a definite integral which evaluates to log⁡e2\log_e 2.
58 Easy
58 of 791
Application +4 | -1
lim⁡n→∞{1n+1+1n+2+1n+3+…+1n+n},\lim\limits_{{n \rightarrow \infty}} \left\{\frac{1}{n+1}+\frac{1}{n+2}+\frac{1}{n+3}+\ldots+\frac{1}{n+n}\right\}, is equal to
View answer and explanation
Correct option: C
log⁡e2\log _e 2
Option C (log⁡e2\log _e 2) is correct because the limit of the sum can be expressed as a definite integral.
59 Easy
59 of 791
Analyzing +4 | -1
If In=∫0πexsin⁡nxdxI_n=\int_0^\pi e^x \sin ^n x d x, then I3I1\frac{I_3}{I_1} is equal to
View answer and explanation
Correct option: A
35\frac{3}{5}
Option A (3/5) is correct because the ratio I3/I1 is derived using integration by parts and recurrence relation.
60 Easy
60 of 791
Application +4 | -1
The value of ∫032π/31+cos⁡2x dx\int\limits_{0}^{32 \pi / 3} \sqrt{1+\cos 2 x} \, d x, is
View answer and explanation
Correct option: B
44−32\frac{44-\sqrt{3}}{\sqrt{2}}
Option B (44−32\frac{44-\sqrt{3}}{\sqrt{2}}) is correct because the integral is evaluated by breaking it into intervals where cos(x) is either positive or negative.
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