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JEE • MATHEMATICS • DEFINITE INTEGRATION

Definite integration practice · Medium

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241 Medium
241 of 596
Analyzing +4 | -1
The integral ∫1e{(xe)2x−(ex)x}log⁡exdx\int\limits_{1}^e\left\{\left(\frac{x}{e}\right)^{2 x}-\left(\frac{e}{x}\right)^x\right\} \log _e x d x is equal to:
View answer and explanation
Correct option: D
32−e−12e2\frac{3}{2}-e-\frac{1}{2 e^2}
Option D (32−e−12e2\frac{3}{2}-e-\frac{1}{2 e^2}) is correct because the integral is solved by using substitution and simplification.
242 Medium
242 of 596
Application +4 | -1
If I(m,n)=∫01tm(1+t)ndtI(m, n)=\int\limits_{0}^1 t^m(1+t)^n d t then the expression for I(m,n)I(m, n) in terms of I(m+1I(m+1, n−1n-1 ) is
View answer and explanation
Correct option: A
2nm+1−nm+1I(m+1,n−1)\frac{2^n}{m+1}-\frac{n}{m+1} I(m+1, n-1)
Option A is correct because it is derived using integration by parts, taking u=(1+t)nu=(1+t)^n and dv=tmdtdv=t^m dt.
243 Medium
243 of 596
Application +4 | -1
The value of ∫23x5−x+xdx\int\limits_{2}^3 \frac{\sqrt{x}}{\sqrt{5-x}+\sqrt{x}} d x is
View answer and explanation
Correct option: D
12\frac{1}{2}
Option D (12\frac{1}{2}) is correct because 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 simplifies the integral to 12\frac{1}{2}.
244 Medium
244 of 596
Analyzing +4 | -1
∫−1/21/2  (cos⁡  x)[log⁡  (1−x1+x)]   dx\int \nolimits_{ - 1/2}^{1/2} \,\,(\cos \,\,x)\left[ {\log \,\,\left( {\frac{{1 - x}}{{1 + x}}} \right)} \right]\,\,\,dx is equal to
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 is evaluated over a symmetric interval.
245 Medium
245 of 596
Application +4 | -1
∫0πdx1−2acos⁡x+a2\int\limits_{0}^\pi \frac{d x}{1-2 a \cos x+a^2} is equal to
View answer and explanation
Correct option: A
(1+a2)π(1−a2)2\frac{\left(1+a^2\right) \pi}{\left(1-a^2\right)^2}
Option A ((1+a2)π(1−a2)2\frac{\left(1+a^2\right) \pi}{\left(1-a^2\right)^2}) is correct because it is the result of evaluating the definite integral using trigonometric substitution and properties.
246 Medium
246 of 596
Analyzing +4 | -1
Area enclosed by the curve 5x2+2y2+6xy+7x+6y+6=05 x^2+2 y^2+6 x y+7 x+6 y+6=0 is (in sq. units)
View answer and explanation
Correct option: A
π2\frac{\pi}{2}
Option A (π2\frac{\pi}{2}) is correct because it represents the area enclosed by the given ellipse.
247 Medium
247 of 596
Application +4 | -1
If a  <  ∫02π  110  +  3  cos⁡  xdx  <  ba\,\, < \,\,\int \nolimits_0^{2\pi } \,\,\frac{1}{{10\,\, + \,\,3\,\,\cos \,\,x}}dx\,\, < \,\,b, then the ordered pair (a,  b)\left( {a,\,\,b} \right) is

View answer and explanation
Correct option: B
(2π13,  2π7)\left( {\frac{{2\pi }}{{13}},\,\,\frac{{2\pi }}{7}} \right)
Option B is correct because the integral is bounded by the minimum and maximum values of the integrand over the given interval, resulting in the interval (2π/13, 2π/7).
248 Medium
248 of 596
Application +4 | -1
∫13  (x−1)  (x−2)  (x−3)  dx\int \nolimits_1^3 \,\,\left( {x - 1} \right)\,\,\left( {x - 2} \right)\,\,\left( {x - 3} \right)\,\,dx is equal to
View answer and explanation
Correct option: D
0\mathrm{0}
Option D (0) is correct because the integral evaluates to zero after expanding the polynomial and applying the limits of integration.
249 Medium
249 of 596
Application +4 | -1
The value of ∫1e37πsin⁡(πln⁡x)xdx\int\limits_{1}^{e^{37}} \frac{\pi \sin (\pi \ln x)}{x} d x is ____.
View answer and explanation
Correct option: A
2
Option A (2) is correct because after substitution, the integral simplifies to the integral of sin(t) from 0 to 37pi, which evaluates to 2.
250 Medium
250 of 596
Application +4 | -1
For 0<a<1,0 < a < 1, the value of the integral ∫0πdx1−2acos⁡x+a2\int\limits_{0}^\pi \frac{d x}{1-2 a \cos x+a^2} is:
View answer and explanation
Correct option: C
π1−a2\frac{\pi}{1-a^2}
Option C is correct because it involves evaluating a definite integral using trigonometric substitution and properties of definite integrals.
251 Medium
251 of 596
Application +4 | -1
∫log⁡π/2log⁡πe2xsec⁡2(13e2x)dx\int\limits_ {\log \sqrt{\pi / 2}}^{\log \sqrt{\pi}} {e}^{2 {x}} \sec ^2\left(\frac{1}{3} {e}^{2 {x}}\right) {dx} is equal to :
View answer and explanation
Correct option: A
3\sqrt{3}
Option A (Only I and II are correct) is correct because statements I and II are true based on the given probabilities, while statement III is false.
252 Medium
252 of 596
Analyzing +4 | -1
∫0π  [cot⁡  x]  dx,  [ . ]  \int \nolimits_0^\pi \,\,[\cot \,\,x]\,\,dx,\,\,\left[ {\,.\,} \right]\,\, denotes the greatest integer function, is equal to
View answer and explanation
Correct option: D
−π2- \frac{\pi }{2}
Option D (-π/2) is correct because using the property ∫0π[cot x] dx = ∫0π[-cot x] dx and [x] + [-x] = -1, the integral simplifies to -π/2.
253 Medium
253 of 596
Application +4 | -1
The value of the integral ∫0πx1+cos⁡2xdx\int\limits_{0}^\pi \frac{x}{1+\cos ^2 x} d x is
View answer and explanation
Correct option: B
π222\frac{\pi^2}{2 \sqrt{2}}
Option B (π222\frac{{\pi^2}}{{2 \sqrt{{2}}}}) is correct because the integral is solved using properties of definite integrals and trigonometric substitutions.
254 Medium
254 of 596
Application +4 | -1
Let f:(0,  ∞)→Rf:\left( {0,\,\,\infty } \right) \to R and F(x)=∫0x  f(x)  dtF\left( x \right) = \int \nolimits_0^x \,\,f\left( x \right)\,\,dt. If F(x2)=x2(1+x)F\left( {{x^2}} \right) = {x^2}\left( {1 + x} \right), then f(4)f\left( 4 \right) equals
View answer and explanation
Correct option: C
4\mathrm{4}
Option C (4) is correct because differentiating F(x2) and solving for f(x) allows us to find f(4) = 4.
255 Medium
255 of 596
Analyzing +4 | -1
If I=∫−11  {[x2]  +  log⁡  (2+x2−x)}dxI = \int \limits_{ - 1}^1 \,\,\left\{ {\left[ {{x^2}} \right]\,\, + \,\,\log \,\,\left( {\frac{{2 + x}}{{2 - x}}} \right)} \right\}dx where [x]\left[ x \right] denotes the greatest integer less than or equal to xx, the II equals
View answer and explanation
Correct option: C
0\mathrm{0}
The correct answer is C (0) because the integral simplifies to the integral of an even function plus the integral of an odd function, and the integral of the odd function over the symmetric interval is zero.

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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 Medium
256 of 596
Understanding +4 | -1
Let f(x)f(x) be a continuous function defined on [0,a][0, a] such that f(a−x)=f(x)f(a-x)=f(x) for all x∈[0,a]x \in[0, a]. If ∫0a/2f(x)dx=α\int_0^{a / 2} f(x) d x=\alpha, then ∫0af(x)dx\int_0^a f(x) d x is equal to
View answer and explanation
Correct option: B
2α2 \alpha
Option B (2α2 \alpha) is correct because using the property ∫0af(x)dx=∫0a/2[f(x)+f(a−x)]dx\int_0^a f(x) dx = \int_0^{a/2} [f(x) + f(a-x)] dx and the given condition f(a−x)=f(x)f(a-x)=f(x), we get ∫0af(x)dx=2∫0a/2f(x)dx=2α\int_0^a f(x) dx = 2 \int_0^{a/2} f(x) dx = 2\alpha.
257 Medium
257 of 596
Application +4 | -1
∫−11xtan⁡−1xdx\int\limits_{{-1}}^1 x \tan ^{-1} x d x equals
View answer and explanation
Correct option: A
(π2−1)\left(\frac{\pi}{2}-1\right)
Option A ((π2−1)\left(\frac{\pi}{2}-1\right)) is correct because the integral of xtan⁡−1xx \tan^{-1} x from -1 to 1 is equal to π2−1\frac{\pi}{2}-1.
258 Medium
258 of 596
Analyzing +4 | -1
If ∫ab  xnxn+  (16−x)n  dx=6\int \limits_a^b \,\,\frac{{{x^n}}}{{{x^n} + \,\,{{\left( {16 - x} \right)}^n}}}\,\,dx = 6, then
View answer and explanation
Correct option: B
a=2,  b=14,  n∈Ra = 2,\,\,b = 14,\,\,n \in R
Option B (a=2, b=14, n∈Ra = 2,\,b = 14,\,n \in R) is correct because it satisfies the property of definite integrals where the integral's value depends on the limits of integration, and a+b=16a+b = 16 and b−a2=6\frac{b-a}{2} = 6.
259 Medium
259 of 596
Application +4 | -1
If AA is the area lying between the curve y=sin⁡xy = \sin x and x−x - axis between x=0x = 0 and x=π/2x = \pi /2. Area of the region between the curve y=sin⁡2xy = \sin 2x and xx-axis in the same interval is given by
View answer and explanation
Correct option: B
A
Option A (A2\frac{A}{2}) is incorrect. The correct answer is B (A). The provided solution incorrectly states the answer is A/2. The correct area is A.
260 Medium
260 of 596
Application +4 | -1
If [x][x] represents integral part of xx, then the value of the integral ∫−1/21/2([x]+log⁡e(1+x1−x))dx\int\limits_{{-1 / 2}}^{1 / 2}\left([x]+\log _e\left(\frac{1+x}{1-x}\right)\right) d x is
View answer and explanation
Correct option: A
−12\frac{-1}{2}
Option A is correct because the integral of [x] from -1/2 to 1/2 is -1/2, and the integral of the log term is zero due to it being an odd function.
261 Medium
261 of 596
Application +4 | -1
If ∫sin⁡θcos⁡θf(xtan⁡θ)dx(θ≠nπ2,n∈I)\int_{\sin \theta}^{\cos \theta} f(x \tan \theta) d x\left(\theta \neq \frac{n \pi}{2}, n \in I\right) is equal to
View answer and explanation
Correct option: A
−cos⁡θ∫1tan⁡θf(xsin⁡θ)dx-\cos \theta \int_1^{\tan \theta} f(x \sin \theta) d x
Option A is correct because by substituting xtan⁡θ=zsin⁡θx \tan \theta = z \sin \theta and changing the limits of integration, the integral can be simplified to the form given in option A.
262 Medium
262 of 596
Application +4 | -1
If f(a+b−x)=f(x)f(a+b-x)=f(x), then ∫abxf(x)dx=\int\limits_{{a}}^{b} x f(x) d x=
View answer and explanation
Correct option: C
a+b2∫abf(x)dx\frac{a+b}{2} \int\limits_{{a}}^{b} f(x) d x
Option C (a+b2∫abf(x)dx\frac{a+b}{2} \int\limits_{{a}}^{b} f(x) d x) is correct because using the property f(a+b−x)=f(x)f(a+b-x) = f(x) simplifies the integral to this form.
263 Medium
263 of 596
Application +4 | -1
∫dxcos⁡x−sin⁡x\int \frac{d x}{\cos x-\sin x} is equal to
View answer and explanation
Correct option: A
12log⁡∣tan⁡(x2+3π8)∣+C\frac{1}{\sqrt{2}} \log \left|\tan \left(\frac{x}{2}+\frac{3 \pi}{8}\right)\right|+C
Option A (12log⁡∣tan⁡(x2+3π8)∣+C\frac{1}{\sqrt{2}} \log |\tan (\frac{x}{2}+\frac{3 \pi}{8})|+C) is correct because the integral simplifies to a secant integral, which can be solved using the tangent half-angle formula.
264 Medium
264 of 596
Application +4 | -1
∫−ππ∣π−∣x∣∣dx\int\limits_{{-\pi}}^\pi|\pi-| x|| d x is equal to:
View answer and explanation
Correct option: C
π2\pi^2
Option C (π2\pi^2) is correct because the integral can be simplified using the symmetry of the absolute value function.
265 Medium
265 of 596
Application +4 | -1
If [ ⋅ ]\left[ {\, \cdot \,} \right] denotes the greatest integer function and f(x)={3[x]  −  5  ∣ x ∣x,    x≠02                            ,  x=0f\left( x \right) = \left\{ {\begin{array}{cccccccccccccccccccc} {3\left[ x \right]\,\, - \,\,\frac{{5\,\,\left| {\,x\,} \right|}}{x},\,\,\;x \ne 0}\\ {2\;\;\;\;\;\;\;\;\;\;\;\;\;\;,\,\,x = 0} \end{array}} \right. then ∫−3/22  f(x)dx\int \limits_{ - 3/2}^2 \,\,f\left( x \right)dx is equal to
View answer and explanation
Correct option: A
−112- \frac{{11}}{2}
Option A (-11/2) is correct because the definite integral is evaluated by splitting the integral based on the greatest integer function and absolute value.
266 Medium
266 of 596
Application +4 | -1
The value of ∫−13  [tan⁡−1  (xx2+  1)  +  tan⁡−1  (x2+1x)]    dx\int \nolimits_{ - 1}^3 \,\,\left[ {{{\tan }^{ - 1}}\,\,\left( {\frac{x}{{{x^2} + \,\,1}}} \right)\,\, + \,\,{{\tan }^{ - 1}}\,\,\left( {\frac{{{x^2} + 1}}{x}} \right)} \right]\;\,\,dx is
View answer and explanation
Correct option: A
2 π2\,\mathrm{\pi }
Option A (2π2\pi) is correct because the expression inside the integral simplifies to π/2\pi/2, and the integral of a constant is straightforward.
267 Medium
267 of 596
Application +4 | -1
The value of the integral ∫0π/2  ∣ sin⁡  x  −  cos⁡  x ∣  dx\int \limits_0^{\pi /2} \,\,\left| {\,\sin \,\,x\,\, - \,\,\cos \,\,x\,} \right|\,\,dx, is
View answer and explanation
Correct option: B
2  (2−1)2\,\,\left( {\sqrt 2 - 1} \right)
Option B (2  (2−1)2\,\,\left( {\sqrt 2 - 1} \right)) is correct because the integral of the absolute value function splits into two integrals, and the result is 22−22\sqrt{2} - 2.
268 Medium
268 of 596
Application +4 | -1
If f(x)=Asin⁡πx2+B,f′(1/2)=2f(x)=A \sin \frac{\pi x}{2}+B, f^{\prime}(1 / 2)=\sqrt{2} and ∫01f(x)dx=2Aπ\int\limits_{0}^1 f(x) \mathrm{d} x=\frac{2 A}{\pi} then the constants AA and BB are respectively
View answer and explanation
Correct option: D
4/π,04 / \pi, 0
Option D (4/π\pi, 0) is correct because solving the given equations for A and B yields these values.
269 Medium
269 of 596
Application +4 | -1
The least integer greater than ∫0100{x}dx\int_0^{100}\{\sqrt{x}\} d x is (where {.} is fractional part function)
View answer and explanation
Correct option: C
52
Option C (52) is correct because the integral evaluates to a value between 51 and 52, and 52 is the least integer greater than that value.
270 Medium
270 of 596
Application +4 | -1
If ∫log⁡2x  du(eu−1)1/2=π6\int \nolimits_{\log 2}^x \,\,\frac{{du}}{{{{\left( {{e^u} - 1} \right)}^{1/2}}}} = \frac{\pi }{6}, then ex{e^x} is equal to
View answer and explanation
Correct option: C
4\mathrm{4}
Option C (4) is correct because it is the result of solving the definite integral and then solving for e^x.
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