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JEE • MATHEMATICS • DEFINITE INTEGRATION

Definite integration practice · Mixed difficulty

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241 Easy
241 of 995
Application +4 | -1
The value of ∫−23  ∣ 1−x2 ∣  dx  is\int \nolimits_{ - 2}^3 \,\,\left| {\,1 - {x^2}\,} \right|\,\,dx\,\,\mathrm{is}
View answer and explanation
Correct option: A
283\frac{{28}}{3}
Option A (283\frac{{28}}{3}) is correct because it is the result of correctly evaluating the definite integral of the absolute value function over the given interval.
242 Easy
242 of 995
Understanding +4 | -1
If f(a+x)=f(x)f\left( {a + x} \right) = f\left( x \right), then ∫0na f  (x)  dx\int \limits_0^{na} \,f\,\,\left( x \right)\,\,dx is equal to (n∈N)\left( {n \in N} \right)
View answer and explanation
Correct option: B
n ∫0a f(x)  dxn\,\int \limits_0^a \,f\left( x \right)\,\,dx
Option B (n∫0a,f(x),,dxn\int \limits_0^a \\,f\left( x \right)\\,,dx) is correct because the integral over nana can be split into nn integrals over the interval [0,a][0, a] due to the periodicity of f(x)f(x).
243 Easy
243 of 995
Application +4 | -1
The value of ∫−24  ∣x+1∣  dx\int \nolimits_{ - 2}^4 \,\,\left| {x + 1} \right|\,\,dx is equal to
View answer and explanation
Correct option: C
13\mathrm{13}
Option C is correct because the integral of |x+1| from -2 to 4 is calculated by splitting the integral at x = -1, resulting in a value of 13.
244 Medium
244 of 995
Application +4 | -1
If b   >  ab\,\,\, > \,\,a, then ∫ab  dx(x−a)  (b−x)\int \nolimits_a^b \,\,\frac{{dx}}{{\sqrt {\left( {x - a} \right)\,\,\left( {b - x} \right)} }} is equal to
View answer and explanation
Correct option: B
π\pi
Option B (π\pi) is correct because it is the value obtained after evaluating the definite integral using trigonometric substitution.
245 Easy
245 of 995
Application +4 | -1
The value of ∫0∞  dx(x2+  4)  (x2+  9)\int \nolimits_0^\infty \,\,\frac{{dx}}{{\left( {{x^2} + \,\,4} \right)\,\,\left( {{x^2} + \,\,9} \right)}} is
View answer and explanation
Correct option: A
π60\frac{\pi }{{60}}
Option A (π60\frac{\pi}{{60}}) is correct because after using partial fractions and integrating, the result is π60\frac{\pi}{{60}}.
246 Medium
246 of 995
Application +4 | -1
The value of integral ∫0π  x    f(sin⁡  x)    dx    is\int \nolimits_0^\pi \,\,x\;\,\,f(\sin \,\,x)\;\,\,dx\,\,\;\mathrm{is}
View answer and explanation
Correct option: B
π ∫0π/2  f  (sin⁡  x)   dx\pi \,\int \nolimits_0^{\pi /2} \,\,f\;(\sin \,\,x)\;\,dx
Option B (π ∫0π/2 f  (sin⁡ x) dx\pi \,\int \nolimits_0^{\pi /2} \,f\;(\sin \,x)\,dx) is correct because it uses the property of definite integrals to simplify the expression.
247 Medium
247 of 995
Application +4 | -1
If area bounded by the curves y2=4  ax and y=mx{y^2} = 4\;ax\,\mathrm{and}\,y = mx is a2/3,{a^2}/3, then the value of mm is
View answer and explanation
Correct option: A
2
The correct answer is A (2) because it is the value of m that makes the area enclosed by the curves equal to a²/3, which is found by integrating the difference between the curves.
248 Easy
248 of 995
Understanding +4 | -1
∫−aasin⁡xf(cos⁡x)dx=\int\limits_{{-a}}^a \sin x f(\cos x) d x=
View answer and explanation
Correct option: B
0
Option B is correct because sin(x) is an odd function and f(cos(x)) is an even function, making the product an odd function, and the integral of an odd function over a symmetric interval is zero.
249 Medium
249 of 995
Application +4 | -1
The integral ∫dxx2(x4+1)1/4\int \frac{d x}{x^2\left(x^4+1\right)^{1 / 4}} equals
View answer and explanation
Correct option: B
−(x4+1x4)14+c-\left(\frac{x^4+1}{x^4}\right)^{\frac{1}{4}}+c
Option B (−(x4+1x4)14+c-\left(\frac{x^4+1}{x^4}\right)^{\frac{1}{4}}+c) is correct because after substitution and integration, the integral evaluates to this expression.
250 Easy
250 of 995
Analyzing +4 | -1
The value of ∫02π[sin⁡2x(1+cos⁡3x)]dx,\int\limits_0^{2\pi } {\left[ {\sin 2x(1 + \cos 3x)} \right]dx,} where [t][t] denotes the greatest integer function, is
View answer and explanation
Correct option: A
−π- \pi
Option A ( −π\ - \pi) is correct because the integral simplifies to ∫02π[sin⁡2x(1+cos⁡3x)]dx=−π\int_0^{2\pi} [\sin 2x (1 + \cos 3x)] dx = -\pi after using the property [x]+[−x]=−1[x] + [-x] = -1 for non-integer xx.
251 Easy
251 of 995
Application +4 | -1
The function f(x),f(x), that satisfies the condition f(x)=x+∫0π/2sin⁡x.cos⁡y  f(y)dy,f(x) = x + \int\limits_0^{\pi /2} {\sin x.\cos y\,\,f(y)dy,} is
View answer and explanation
Correct option: D
x+(π−2)sin⁡xx + (\pi - 2)\sin x
Option D (x+(π−2)sin⁡xx + (\pi - 2)\sin x) is the correct function because it satisfies the given integral equation.
252 Easy
252 of 995
Application +4 | -1
If f(x)={x2,0≤x≤1x,1≤x≤2f(x)=\left\{\begin{array}{ll}x^2, & 0 \leq x \leq 1 \\ \sqrt{x}, & 1 \leq x \leq 2\end{array}\right. then ∫02f(x)dx=\int\limits_{0}^2 f(x) d x=
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 given options.
253 Easy
253 of 995
Analyzing +4 | -1
∫−11  (ex3+  e−x3)(ex−e−x)dx  is  equal  to\int \nolimits_{ - 1}^1 \,\,\left( {{e^x}^3 + \,\,{e^{ - x}}^3} \right)\left( {{e^x} - {e^{ - x}}} \right)dx\,\,\mathrm{is\;equal\;to}
View answer and explanation
Correct option: D
0\mathrm{0}
Option D (0) is correct because the integrand is an odd function, and the integral of an odd function over a symmetric interval is zero.
254 Medium
254 of 995
Application +4 | -1
Suppose ff is such that f  (−x)=− f (x)f\,\,\left( { - x} \right) = - \,f\,\left( x \right) for every real xx and ∫01  f(x)  dx=5\int \nolimits_0^1 \,\,f\left( x \right)\,\,dx = 5, then ∫−10    f(t)  dt\int \nolimits_{ - 1}^0 \,\,\,\,f\left( t \right)\,\,dt is equal to
View answer and explanation
Correct option: D
− 5- \,5
Option D (-5) is correct because f(x) is an odd function, and the integral from -1 to 0 is the negative of the integral from 0 to 1.
255 Easy
255 of 995
Application +4 | -1
The value of the sum ∑n=1100∫n−1nex−[x]dx\sum\limits_{{n=1}}^{100} \int\limits_{{n-1}}^n e^{x-[x]} d x, where [⋅][\cdot] denotes the greatest integer function, is
View answer and explanation
Correct option: B
100(e−1)100(e-1)
Option B (100(e−1)100(e-1)) is correct because the sum of integrals can be combined into a single integral, which simplifies due to the periodic nature of the function.

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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
256 Easy
256 of 995
Application +4 | -1
The value of ∫016  π/3  ∣ sin⁡  x ∣  dx\int \limits_0^{16\;\pi /3} \,\,\left| {\,\sin \,\,x\,} \right|\,\,dx, is
View answer and explanation
Correct option: B
21 / 2\mathrm{21}\,\mathrm{/}\,\mathrm{2}
Option B (21 / 2\mathrm{21}\,\mathrm{/}\,\mathrm{2}) is correct because it involves integrating the absolute value of sine over multiple periods.
257 Easy
257 of 995
Application +4 | -1
∫1/22  ∣ log⁡10  x ∣dx\int \limits_{1/2}^2 \,\,\left| {\,{{\log }_{10}}\,\,x\,} \right|dx is equals to
View answer and explanation
Correct option: B
12  log⁡10  (8/e)\frac{1}{2}\,\,{\log _{10}}\,\,\left( {8/e} \right)
Option B (12 log⁡10  (8/e)\frac{1}{2}\,{\log _{10}}\,\,\left( {8/e} \right)) is correct because it is the result of evaluating the definite integral after splitting it based on the absolute value of the logarithm.
258 Easy
258 of 995
Understanding +4 | -1
To find the numerical value of ∫−22(px2+qx+s)dx\int\limits_{{-2}}^2\left(p x^2+q x+s\right) d x, it is necessary to know the values of constants
View answer and explanation
Correct option: D
pp and ss
Option D is correct because the integral depends on both p and s, as the integral of px² and s are non-zero over the given interval.
259 Medium
259 of 995
Application +4 | -1
The integral ∫7π/47π/3tan⁡2xdx\int\limits_{{7 \pi / 4}}^{7 \pi / 3} \sqrt{\tan ^2 x} d x is equal to :
View answer and explanation
Correct option: D
log⁡2\log \sqrt{2}
Option D (log⁡2\log \sqrt{2}) is correct because after evaluating the integral of tan⁡2x\sqrt{\tan^2 x} from 7π4\frac{7\pi}{4} to 7π3\frac{7\pi}{3}, we obtain log⁡2\log \sqrt{2}.
260 Medium
260 of 995
Application +4 | -1
∫−π/2π/2  sin⁡4  x  cos⁡6  x  dx\int \nolimits_{ - \pi /2}^{\pi /2} \,\,{\sin ^4}\,\,x\,\,{\cos ^6}\,\,x\,\,dx is equal to
View answer and explanation
Correct option: C
3 π256\frac{{3\,\mathrm{\pi }}}{{256}}
Option C (3,π256\frac{{3\\,\mathrm{\pi }}}{{256}}) is correct because the integral of the even function is solved using the Gamma function.
261 Medium
261 of 995
Application +4 | -1
∫01∣sin⁡2πx∣dx=\int\limits_{0}^1|\sin 2 \pi x| d x=
View answer and explanation
Correct option: B
2π\frac{2}{\pi}
Option B (2π\frac{2}{\pi}) is correct because after substitution and using the properties of the sine function, the integral evaluates to 2π\frac{2}{\pi}.
262 Easy
262 of 995
Application +4 | -1
The area bounded by the xx-axis, the curve y=f(x)y=f(x) and the lines x=1,x=bx=1, x=b is equal to b2+1−2\sqrt{b^2+1}-\sqrt{2} for all b>1b>1, then f(x)f(x), is
View answer and explanation
Correct option: D
x/x2+1x / \sqrt{x^2+1}
Option D (x/x2+1x / \sqrt{x^2+1}) is correct because it is obtained by differentiating the given equation with respect to bb using the Fundamental Theorem of Calculus.
263 Medium
263 of 995
Analyzing +4 | -1
If f(x)=  ∣ sec⁡  xcos⁡  xsec⁡2  x  +  cosec  x  cot⁡  xcos⁡2  xcos⁡2  xcosec2  x1cos⁡2  xcos⁡2  x ∣f\left( x \right) = \,\,\left| {\,\begin{array}{cccccccccccccccccccc} {\sec \,\,x}&{\cos \,\,x}&{{{\sec }^2}\,\,x\,\, + \,\,\mathrm{cosec}\,\,x\,\,\cot \,\,x}\\ {{{\cos }^2}\,\,x}&{{{\cos }^2}\,\,x}&{\mathrm{cose}{\mathrm{c}^2}\,\,x}\\ 1&{{{\cos }^2}\,\,x}&{{{\cos }^2}\,\,x} \end{array}\,} \right|, then ∫0π/2  f(x)  dx\int \limits_0^{\pi /2} \,\,f\left( x \right)\,\,dx equals
View answer and explanation
Correct option: C
−  (π4  +  815)- \,\,\left( {\frac{\pi }{4}\,\, + \,\,\frac{8}{{15}}} \right)
Option C (−(π4+815)- \left( {\frac{\pi }{{4}}\\ + \frac{{8}}{{15}}} \right)) is correct because after evaluating the determinant and integrating the resulting trigonometric functions, we obtain this value.
264 Easy
264 of 995
Application +4 | -1
∫0π/2dx1+(1/6)sin⁡2x=\int\limits_{0}^{\pi / 2} \frac{d x}{1+(1 / 6) \sin ^2 x}=
View answer and explanation
Correct option: A
π267\frac{\pi}{2} \sqrt{\frac{6}{7}}
Option A (π267\frac{\pi}{2} \sqrt{\frac{6}{7}}) is correct because after applying trigonometric identities and integrating, the result is π267\frac{\pi}{2} \sqrt{\frac{6}{7}}.
265 Easy
265 of 995
Application +4 | -1
Let f(x)f(x) be a differentiable function defined on [0,2],\left[ {0,2} \right], such that f′(x)=f′(2−x)f'(x) = f'(2 - x), for all x∈(0,  2), x \in \left( {0,\,\,2} \right),\, f(0)=1f(0) = 1 and f(2)=e2f(2) = {e^2}. Then, the value of ∫02f(x)dx\int\limits_0^2 {f(x)dx} is
View answer and explanation
Correct option: B
1+e21 + {e^2}
Option B (1+e21 + {e^2}) is the correct value of the integral, obtained using the properties of definite integrals and the given conditions.
266 Easy
266 of 995
Application +4 | -1
If ∫0π/2  sin⁡6  x dx=5 π32\int \nolimits_0^{\pi /2} \,\,{\sin ^6}\,\,x\,dx = \frac{{5\,\pi }}{{32}}, then the value of ∫−ππ  (sin⁡6x  +  cos⁡6  x)    dx\int \nolimits_{ - \pi }^\pi \,\,({\sin ^6}x\,\, + \,\,{\cos ^6}\,\,x)\;\,\,dx is
View answer and explanation
Correct option: D
5 π  /  45\,\mathrm{\pi }\,\,/\,\,4
Option D (5 π / 45\,\mathrm{\pi}\,/\,4) is correct because the integral can be simplified using trigonometric identities and the given value of the integral from 0 to π/2\pi/2.
267 Easy
267 of 995
Application +4 | -1
∫−11  cos⁡  h    x1  +  e2x    dx\int \nolimits_{ - 1}^1 \,\,\frac{{\cos \,\,\mathrm{h}\;\,\,x}}{{1\,\, + \,\,{e^{2x}}}}\,\,\;dx is equal to
View answer and explanation
Correct option: C
e2−12e\frac{{{e^2} - 1}}{{2e}}
Option C (e2−12e\frac{{{e^2} - 1}}{{2e}}) is correct because the integral of the given function is solved using properties of hyperbolic cosine and simplification.
268 Easy
268 of 995
Understanding +4 | -1
If (−1,  2)\left( { - 1,\,\,2} \right) and (2,4)\left( {2,4} \right) are two points on the curve y=f(x)y = f\left( x \right) and if g(x)g\left( x \right) is the gradient of the curve at point (x,y)\left( {x,y} \right), then the value of the integral ∫−12  g  (x)  dx\int \limits_{ - 1}^2 \,\,g\,\,\left( x \right)\,\,dx, is
View answer and explanation
Correct option: A
2\mathrm{2}
Option A (2) is correct because the integral of the gradient g(x)g(x) is the difference in the function's values at the limits of integration, f(2)−f(−1)=4−2=2f(2) - f(-1) = 4 - 2 = 2.
269 Medium
269 of 995
Analyzing +4 | -1
The least value of the function
F(x)=∫0x(3sin⁡u+4cos⁡u)duF(x)=\int_0^x(3 \sin u+4 \cos u) \mathrm{d} u

on the interval (5π/4,4π/3](5 \pi / 4,4 \pi / 3] is
View answer and explanation
Correct option: D
9−432\frac{9-4 \sqrt{3}}{2}
Option D (9−432\frac{9-4 \sqrt{3}}{2}) is correct because the minimum value of the function occurs at x=4π/3x = 4\pi/3 within the given interval, and the function value there is 9−432\frac{9-4 \sqrt{3}}{2}.
270 Easy
270 of 995
Application +4 | -1
The value of ∫0π/22sin⁡x2sin⁡x+2cos⁡xdx\int\limits_{0}^{\pi / 2} \frac{2^{\sin x}}{2^{\sin x}+2^{\cos x}} d x is
View answer and explanation
Correct option: A
π4\frac{\pi}{4}
Option A (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}.
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