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JEE • MATHEMATICS • DEFINITE INTEGRATION

Definite integration practice · Mixed difficulty

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841 Easy
841 of 995
Analyzing +4 | -1
If [.] denotes the greatest integer function, then the value of ∫13[x2−2x−2]dx\int\limits_{1}^3\left[x^2-2 x-2\right] d x is equal to
View answer and explanation
Correct option: A
−1−2−3-1-\sqrt{2}-\sqrt{3}
Option A (−1−2−3-1-\sqrt{2}-\sqrt{3}) is correct because it is found by evaluating the integral using properties of the greatest integer function.
842 Easy
842 of 995
Application +4 | -1
The equal to integral ∫01log⁡(1−x+1+x)dx\int\limits_{0}^1 \log (\sqrt{1-x}+\sqrt{1+x}) \mathrm{d} x is
View answer and explanation
Correct option: B
12(log⁡2−1+π2)\frac{1}{2}\left(\log 2-1+\frac{\pi}{2}\right)
Option A (12(log⁡2−1+π2)\frac{1}{2}(\log 2-1+\frac{\pi}{2})) is correct because after performing integration by parts and simplifying, the integral evaluates to this expression.
843 Easy
843 of 995
Application +4 | -1
If f  (x)=  { ∣ x ∣,∣x−2∣,−1   ≤   x   ≤11  <   x   ≤3f\,\,\left( x \right) = \,\,\left\{ {\,\begin{array}{cccccccccccccccccccc} {\left| {\,x\,} \right|,}\\ {\left| {x - 2} \right|,} \end{array}\begin{array}{cccccccccccccccccccc} { - 1\,\,\, \le \,\,\,x\,\,\, \le 1}\\ {1\,\, < \,\,\,x\,\,\, \le 3} \end{array}} \right. then ∫−13  f  (x)  dx\int \nolimits_{ - 1}^3 \,\,f\,\,\left( x \right)\,\,dx is equal to
View answer and explanation
Correct option: C
2\mathrm{2}
Option D (2) is correct because the integral is evaluated by splitting the integral based on the piecewise function definition and then integrating each part.
844 Easy
844 of 995
Application +4 | -1
Area lying in the first quadrant and bounded by the curve y=x3y=x^3 and the line y=4xy=4 x is
View answer and explanation
Correct option: D
8
The correct answer is D (8) because it represents the area between the curve and the line in the first quadrant, calculated by integrating the difference of the functions.
845 Medium
845 of 995
Application +4 | -1
The value of ∫−ππsin⁡  x    f(cos⁡  x)dx\int \limits_{ - \pi }^\pi \sin \,\,x\;\,\,f\left( {\cos \,\,x} \right)dx is
View answer and explanation
Correct option: D
None of these
Option D (None of these) is correct because the integrand is an odd function, and the integral of an odd function over a symmetric interval is zero.
846 Medium
846 of 995
Analyzing +4 | -1
Let f  f\; be apositive function. Let I1=∫1−kk  xf{x(1−x)},  I2=∫1−kk  f{x(1−x)}{I_1} = \int \nolimits_{1 - k}^k \,\,xf\left\{ {x\left( {1 - x} \right)} \right\},\,\,{I_2} = \int \nolimits_{1 - k}^k \,\,f\left\{ {x\left( {1 - x} \right)} \right\}
dx  dx\; where 2k  −  1>02k\,\, - \,\,1 > 0. Then   I1I2\;\frac{{{I_1}}}{{{I_2}}} is
View answer and explanation
Correct option: C
12\frac{1}{2}
Option C (1/2) is correct because by using the property ∫abf(x)dx = ∫abf(a+b-x)dx, we can show that I1 = 1/2 * I2, thus I1/I2 = 1/2.
847 Medium
847 of 995
Application +4 | -1
∫01  x    dx[x+1−x2]  1−x2\int \nolimits_0^1 \,\,\frac{{x\,\,\;dx}}{{\left[ {x + \sqrt {1 - {x^2}} } \right]\,\,\sqrt {1 - {x^2}} }} is equal to
View answer and explanation
Correct option: C
π4\frac{\pi }{4}
Option C (π4\frac{\pi }{4}) is correct because after trigonometric substitution and using properties of definite integrals, the integral simplifies to π4\frac{\pi}{4}.
848 Easy
848 of 995
Analyzing +4 | -1
Let f(x)=∫  x2 dx(1+x2)  (1+1+x2)f\left( x \right) = \int \,\,\frac{{{x^2}\,dx}}{{\left( {1 + {x^2}} \right)\,\,\left( {1 + \sqrt {1 + {x^2}} } \right)}} and f(0)=0f\left( 0 \right) = 0. Then, f(1)f\left( 1 \right) is
View answer and explanation
Correct option: B
log⁡  (1+2)  −  π4\log \,\,\left( {1 + \sqrt 2 } \right)\,\, - \,\,\frac{\pi }{4}
Option B (log⁡,(1+,2),−,,π4\log \\,\left( {1\\ + \\,\sqrt 2 } \right)\\,\\ - \\,\\,\frac{\pi }{4}) is correct because after solving the integral and applying the given condition f(0)=0f(0) = 0, we find that f(1)=log⁡,(1+,2),−,,π4f(1) = \log \\,\left( {1\\ + \\,\sqrt 2 } \right)\\,\\ - \\,\\,\frac{\pi }{4}.
849 Medium
849 of 995
Analyzing +4 | -1
If f(t)=∫−tt  e−∣x∣2  dx,    then  lim⁡t→∞  f  (t)f\left( t \right) = \int \nolimits_{ - t}^t \,\,\frac{{{e^{ - \left| x \right|}}}}{2}\,\,dx,\;\,\,\mathrm{then}\,\,\lim\limits_{t \to \infty } \,\,f\,\,\left( t \right) is equal to

View answer and explanation
Correct option: A
1\mathrm{1}
Option A (1) is correct because the limit of the integral as t approaches infinity is 1 after evaluating the integral and applying the limit.
850 Easy
850 of 995
Application +4 | -1
If I=∫−ππ  esin⁡xesin⁡x  +  e−sinx  dxI = \int \nolimits_{ - \pi }^\pi \,\,\frac{{{e^{\sin x}}}}{{{e^{\sin x\;}} + \,\,{e^{ - \mathrm{sinx}}}}}\,\,dx, then I equals
View answer and explanation
Correct option: C
π\pi
Option C (π\pi) is correct because using the property of definite integrals and adding the original integral to the transformed integral simplifies the expression to π\pi.
851 Easy
851 of 995
Application +4 | -1
∫−11  ∣1−x∣  dx\int \nolimits_{ - 1}^1 \,\,\left| {1 - x} \right|\,\,dx is equal to
View answer and explanation
Correct option: C
2\mathrm{2}
Option C (2) is correct because the absolute value can be removed by considering the interval and then integrating the resulting expression.
852 Easy
852 of 995
Application +4 | -1
lim⁡n→∞∑r=1n1ner/n\lim \limits_{{n \rightarrow \infty}} \sum\limits_{{r=1}}^n \frac{1}{n} e^{r / n}, is
View answer and explanation
Correct option: B
e−1e-1
Option B (e−1e-1) is correct because the limit of the Riemann sum is equivalent to a definite integral.
853 Medium
853 of 995
Analyzing +4 | -1
The value of the integral ∫−π/4π/4  sin⁡−4 x dx\int \limits_{ - \pi /4}^{\pi /4} \,\,{\sin ^{ - 4}}\,x\,dx, is
View answer and explanation
Correct option: A
−83- \frac{8}{3}
The correct answer is A (−83- \frac{8}{3}) because the integral of the given function over the specified limits is calculated to be -8/3.
854 Medium
854 of 995
Application +4 | -1
The area bounded by the curves y=f(x)y = f\left( x \right), the xx-axis and the ordinates x=1x = 1 and x=bx = b is (b−1)sin⁡(3b+4)\left( {b - 1} \right)\sin \left( {3b + 4} \right). Then, f(x)f\left( x \right) is
View answer and explanation
Correct option: C
sin⁡(3x+4)+3(x−1)cos⁡(3x+4)\sin \left( {3x + 4} \right) + 3\left( {x - 1} \right)\cos \left( {3x + 4} \right)
Option C (sin(3x + 4) + 3(x - 1)cos(3x + 4)) is correct because it is the derivative of the given area function, representing f(x).
855 Easy
855 of 995
Application +4 | -1
Let f:R→Rf: R \rightarrow R be a function defined by f(x)={[x],x≤20,x>2f(x)=\left\{\begin{array}{r}{[x], x \leq 2} \\ 0, x > 2\end{array}\right., where [x][x] is the greatest integer less than or equal to xx. If I=∫−12xf(x2)2+f(x+1)dxI=\int\limits_{{-1}}^2 \frac{x f\left(x^2\right)}{2+f(x+1)} d x, then the value, is
View answer and explanation
Correct option: A
14\frac{1}{4}
Option A (14\frac{1}{4}) is correct because the integral is evaluated by splitting the integration range and using the definition of the greatest integer 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
856 Easy
856 of 995
Application +4 | -1
The expression ∫0n[x]dx∫0n{x}dx\frac{\int\limits_{0}^n[x] d x}{\int\limits_{0}^n\{x\} d x}, where [x][x] and {x}\{x\} are integral and fractional parts of xx and n∈Nn \in N is equal to
View answer and explanation
Correct option: D
n−1n-1
Option D (n−1n-1) is correct because the integral of the integer part divided by the integral of the fractional part simplifies to n−1n-1.
857 Easy
857 of 995
Application +4 | -1
The area of the region bounded by y=x−1y=x-1 and x=3−y2x=3-y^2 is
View answer and explanation
Correct option: C
92\frac{9}{2}
Option C (9/2) is correct because it is the area calculated by integrating the difference between the two functions with respect to y between the intersection points.
858 Easy
858 of 995
Analyzing +4 | -1
∫05cos⁡(π(x−[x2]))dx\int\limits_0^5 \cos \left(\pi\left(x-\left[\frac{x}{2}\right]\right)\right) d x, where [t][t] denotes greatest integer less than or equal to tt, is equal to
View answer and explanation
Correct option: D
0
The correct answer is D (0) because the integral evaluates to zero after splitting the interval and simplifying the cosine function.
859 Easy
859 of 995
Application +4 | -1
∫02πex/2sin⁡(x2+π4)dx=\int\limits_{0}^{2 \pi} e^{x / 2} \sin \left(\frac{x}{2}+\frac{\pi}{4}\right) d x=
View answer and explanation
Correct option: C
0
Option C (0) is correct because the integral simplifies to an expression that evaluates to zero at both limits of integration.
860 Medium
860 of 995
Application +4 | -1
If aa is fixed real number such that f(a−x)  +  f(a+x)=0f\left( {a - x} \right)\,\, + \,\,f\left( {a + x} \right) = 0, then ∫02a  f(x)  dx=\int \limits_0^{2a} \,\,f\left( x \right)\,\,dx =
View answer and explanation
Correct option: B
0\mathrm{0}
Option B (0) is correct because the integral of the function over the interval [0, 2a] is zero due to the property f(a−x)+f(a+x)=0f(a-x) + f(a+x) = 0, which implies f(2a−x)=−f(x)f(2a-x) = -f(x).
861 Easy
861 of 995
Application +4 | -1
If ∫0a  f(2a  −  x)  dx=m    and  ∫0a  f(x)dx=n\int \nolimits_0^a \,\,f\left( {2a\,\, - \,\,x} \right)\,\,dx = m\;\,\,\mathrm{and}\,\,\int \nolimits_0^a \,\,f\left( x \right)dx = n, then ∫02a  f(x)  dx\int \nolimits_0^{2a} \,\,f\left( x \right)\,\,dx is equal to
View answer and explanation
Correct option: D
m  +  nm\,\, + \,\,n
Option D (m+nm\\ + \\ n) is correct because ∫02af(x)dx\int_0^{2a} f(x) dx can be split into two integrals, and using the given information, it equals m+nm\\ + \\ n.
862 Easy
862 of 995
Application +4 | -1
Let ddx(F(x))=esin⁡xx,x>0\frac{d}{d x}(F(x))=\frac{e^{\sin x}}{x}, x > 0
If ∫142esin⁡x2xdx=F(k)−F(1)\int\limits_{1}^4 \frac{2 e^{\sin x^2}}{x} d x=F(k)-F(1), then one of the possible values of kk, is
View answer and explanation
Correct option: C
16
Option C is correct. By using substitution and the given derivative, the integral simplifies to F(16) - F(1), thus k=16.
863 Medium
863 of 995
Application +4 | -1
If [t] denotes the greatest integer ≤t\leq t, then the value of 3(e−1)e∫12x2e[x]+[x3]dx\frac{3(e-1)}{e} \int\limits_{1}^2 x^2 e^{[x]+\left[x^3\right]} d x is
View answer and explanation
Correct option: D
e8−ee^8-e
Option D (e8−ee^8-e) is correct because it is the result of evaluating the definite integral after applying the greatest integer function and simplifying.
864 Easy
864 of 995
Application +4 | -1
The value of ∫01  x4+  1x2+  1  dx\int \nolimits_0^1 \,\,\frac{{{x^4} + \,\,1}}{{{x^2} + \,\,1}}\,\,dx is
View answer and explanation
Correct option: D
16  (3π  −  4 )\frac{1}{6}\,\,\left( {3\pi \,\, - \,\,4\,} \right)
Option D (\frac{1}{6}\\,,\left( {3\pi \\, - \\,4\} \right)\frac{1}{6}\\,,\left( {3\pi \\, - \\,4\} \right)) is correct because after dividing and integrating, the result is \frac{1}{6}\\,,\left( {3\pi \\, - \\,4\} \right)\frac{1}{6}\\,,\left( {3\pi \\, - \\,4\} \right).
865 Medium
865 of 995
Application +4 | -1
The value of ∫0π  15  +  3  cos⁡  x  dx\int \limits_0^\pi \,\,\frac{1}{{5\,\, + \,\,3\,\,\cos \,\,x}}\,\,dx, is
View answer and explanation
Correct option: C
π  /  4\pi \,\,/\,\,4
Option C (π/4\pi \\/\\4) is correct because the definite integral is evaluated using trigonometric substitution and properties of definite integrals.
866 Easy
866 of 995
Application +4 | -1
The value of : ∫π/65π/64−4sin⁡2t dt\int\limits_{{\pi / 6}}^{5 \pi / 6} \sqrt{4-4 \sin ^2 t} \,d t, is
View answer and explanation
Correct option: B
2
Option B (2) is correct because the integral of the given expression is calculated by splitting the integral based on the absolute value and cosine function's sign.
867 Easy
867 of 995
Analyzing +4 | -1
If I1=∫1−kk  x  sin⁡  {x(1−x)}  dx{I_1} = \int \limits_{1 - k}^k \,\,x\,\,\sin \,\,\left\{ {x\left( {1 - x} \right)} \right\}\,\,dx and I2=∫1−kk  sin⁡  {x(1−x)}  dx{I_2} = \int \limits_{1 - k}^k \,\,\sin \,\,\left\{ {x\left( {1 - x} \right)} \right\}\,\,dx, then
View answer and explanation
Correct option: B
2 I1=I22\,{I_1} = {I_2}
Option B (2 I1=I22\,{I_1} = {I_2}) is correct because by using the property of definite integrals and algebraic manipulation, we can derive this relationship.
868 Medium
868 of 995
Application +4 | -1
∫1x  log⁡  (x2)x    dx\int \nolimits_1^x \,\,\frac{{\log \,\,\left( {{x^2}} \right)}}{x}\;\,\,dx is equal to
View answer and explanation
Correct option: A
(log⁡  x)2{(\log \,\,x)^2}
Option A ((log⁡  x)2{(\log \,\,x)^2}) is correct because the integral of the given expression simplifies to this result after substitution.
869 Medium
869 of 995
Application +4 | -1
∫dxcos⁡x+3sin⁡x\int \frac{d x}{\cos x+\sqrt{3} \sin x} equals
View answer and explanation
Correct option: C
12log⁡tan⁡(x2+π12)+C\frac{1}{2} \log \tan \left(\frac{x}{2}+\frac{\pi}{12}\right)+C
Option C (12log⁡tan⁡(x2+π12)+C\frac{1}{2} \log \tan (\frac{x}{2}+\frac{\pi}{12})+C) is correct because the integral can be transformed into a cosecant integral and then solved using the tangent half-angle formula.
870 Easy
870 of 995
Application +4 | -1
The value of ∫0∞  dx(a2+x2)\int \nolimits_0^\infty \,\,\frac{{dx}}{{\left( {{a^2} + {x^2}} \right)}} is equal to
View answer and explanation
Correct option: B
π2 a\frac{\pi }{{2\,a}}
Option B (π2,a\frac{\pi }{{2\\,a}}) is correct because the integral evaluates to (1/a)arctan(x/a) from 0 to infinity, which simplifies to pi/(2a).
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