Skip to main content
Back to DEFINITE INTEGRATION

Your Progress

0 / 791 Answered
0
Correct
0
Incorrect
0%
Accuracy
JEE • MATHEMATICS • DEFINITE INTEGRATION

Definite integration practice · Easy

Choose an answer to check it, or open “View answer” to study the explanation. Sign in to save progress to your account.

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.

Take a chapter quiz →
Showing 571-600 of 791 questions

Preparing practice and saved progress…

571 Easy
571 of 791
Application +4 | -1
Evaluate ∫0π/2  sin⁡  xsin⁡  x  +  cos⁡  x  dx  is\int \nolimits_0^{\pi /2} \,\,\frac{{\sqrt {\sin \,\,x} }}{{\sqrt {\sin \,\,x\,\, + \,\,} \sqrt {\cos \,\,x} }}\,\,dx\,\,\mathrm{is} is〗
View answer and explanation
Correct option: A
π4\frac{\pi }{4}
Option A (π4\frac{\pi}{4}) is correct because using the property of definite integrals ∫0af(x)dx=∫0af(a−x)dx\int \nolimits_0^a f(x) dx = \int \nolimits_0^a f(a-x) dx simplifies the integral.
572 Easy
572 of 791
Application +4 | -1
The value of lim⁡x→0∫0xcos⁡t2 dtx\lim \limits_{{x \rightarrow 0}} \frac{\int\limits_{0}^x \cos t^2 \mathrm{~d} t}{x} is
View answer and explanation
Correct option: A
1
Option A (1) is correct because L'Hopital's rule can be applied to evaluate the limit of the indeterminate form 0/0.
573 Easy
573 of 791
Application +4 | -1
The value of the integral ∫αβdx(x−α)(β−x)\int\limits_ \alpha^\beta \frac{d x}{\sqrt{(x-\alpha)(\beta-x)}} for β>α\beta>\alpha, is
View answer and explanation
Correct option: D
π\pi.
Option D (π\pi) is correct because the integral represents the area under a semicircle, which evaluates to π\pi.
574 Easy
574 of 791
Application +4 | -1
If f(x)f(x) is differentiable and ∫0t2xf(x)dx=25t5\int_0^{t^2} x f(x) d x=\frac{2}{5} t^5, then f(425)f\left(\frac{4}{25}\right) equals
View answer and explanation
Correct option: A
2/52 / 5
Option A (2/5) is correct because differentiating both sides of the equation using Leibniz's rule and solving for f(4/25) leads to the answer 2/5.
575 Easy
575 of 791
Analyzing +4 | -1
If ∫0100 πsin⁡2xe(xπ−[xπ]) dx=απ31+4π2,  α∈R\int\limits_0^{100\,\pi } {\frac{{{{\sin }^2}x}}{{{e^{\left( {\frac{x}{\pi } - \left[ {\frac{x}{\pi }} \right]} \right)}}}}} \,dx = \frac{{\alpha {\pi ^3}}}{{1 + 4{\pi ^2}}},\,\,\alpha \in R, where [x][x] is the greatest integer less than or equal to xx then the value of α\alpha is
View answer and explanation
Correct option: A
200 (1−e−1)200\,(1 - {e^{ - 1}})
Option A (200 (1−e−1)200\,(1 - {e^{ - 1}})) is correct because the integral can be simplified using the periodicity of the sine function and then evaluated, leading to the given result.
576 Easy
576 of 791
Analyzing +4 | -1
If ∫0xf(t)dt=x+∫x1tf(t)dt\int\limits_{0}^x f(t) d t=x+\int_x^1 t f(t) d t then f(1)f(1) is
View answer and explanation
Correct option: A
12\frac{1}{2}
Option A (12\frac{{1}}{{2}}) is correct because by differentiating the given equation and substituting x=1, we find that f(1) equals 12\frac{{1}}{{2}}.
577 Easy
577 of 791
Analyzing +4 | -1
The value of ∫−11log⁡(x+1+x2)x+log⁡(x+1+x2)f(x)dx−∫−11log⁡(x+1+x2)x+log⁡(x+1+x2)f(−x)dx,\int_{ - 1}^1 {\frac{{\log \left( {x + \sqrt {1 + {x^2}} } \right)}}{{x + \log \left( {x + \sqrt {1 + {x^2}} } \right)}}} f(x)dx - \int_{ - 1}^1 {\frac{{\log \left( {x + \sqrt {1 + {x^2}} } \right)}}{{x + \log \left( {x + \sqrt {1 + {x^2}} } \right)}}} f( - x)dx, is
View answer and explanation
Correct option: A
0
Option A (0) is correct because the integrand is an odd function integrated over a symmetric interval.
578 Easy
578 of 791
Application +4 | -1
The value of integral ∫−24  x  [x]  dx\int \limits_{ - 2}^4 \,\,x\,\,\left[ x \right]\,\,dx, is
View answer and explanation
Correct option: A
41  / 2\mathrm{41}\,\,\mathrm{/}\,\mathrm{2}
Option A (41/2) is correct because the integral is evaluated by splitting it into intervals where the greatest integer function is constant, and then summing the results.
579 Easy
579 of 791
Application +4 | -1
The value of the definite integral ∫π245π24dx1+tan⁡2x3\int\limits_{\frac{\pi }{{24}}}^{\frac{{5\pi }}{{24}}} {\frac{{dx}}{{1 + \sqrt[3]{{\tan 2x}}}}} is
View answer and explanation
Correct option: C
π12\frac{\pi }{{12}}
Option C (π12\frac{π }{{12}}) is correct because it is the value of the definite integral after applying the property of definite integrals and simplifying.
580 Easy
580 of 791
Application +4 | -1
The equation of the tangent to the curve
y=∫x4x6dt1+t2 at x=1 is y=\int\limits_{{x^4}}^{x^6} \frac{\mathrm{d} t}{\sqrt{1+t^2}} \text { at } x=1 \text { is }
View answer and explanation
Correct option: D
None of these
Option C (2) is correct because the distance between the parallel lines is calculated by rewriting the equation and using the formula for the distance between parallel lines.
581 Easy
581 of 791
Analyzing +4 | -1
The integral ∫0ag(x)f(x)+f(a−x)dx\int\limits_{0}^a \frac{g(x)}{f(x)+f(a-x)} d x vanishes, if
View answer and explanation
Correct option: C
g(x)=−g(a−x)g(x)=-g(a-x)
Option A (periodic with period TT) is correct because the integral of an odd periodic function over a period is zero, making the resulting function also periodic with period T.
582 Easy
582 of 791
Analyzing +4 | -1
The value ∫−22{pln⁡(1+x1−x)+qln⁡(1−x1+x)−2+r}dx\int_{-2}^2\left\{p \ln \left(\frac{1+x}{1-x}\right)+q \ln \left(\frac{1-x}{1+x}\right)^{-2}+r\right\} d x depends on the value of
View answer and explanation
Correct option: C
rr
Option C (rr) is correct because the integral depends only on the value of rr after simplifying the expression due to the odd functions.
583 Easy
583 of 791
Application +4 | -1
The value of ∫0αdx1−cos⁡αcos⁡x(0<α<π/2)\int\limits_{0}^\alpha \frac{\mathrm{d} x}{1-\cos \alpha \cos x}(0 < \alpha < \pi / 2) is
View answer and explanation
Correct option: A
πsin⁡α\frac{\pi}{\sin \alpha}
Option A (πsin⁡α\frac{\pi}{\sin \alpha}) is correct because after substitution and integration, the result matches this expression.
584 Easy
584 of 791
Analyzing +4 | -1
The value of the integral ∫−1/21/2cos⁡xlog⁡(1+x1−x)dx\int\limits_{{-1 / 2}}^{1 / 2} \cos x \log \left(\frac{1+x}{1-x}\right) d x, is
View answer and explanation
Correct option: A
0
Option A (0) is correct because the integrand is an odd function, and the integral is evaluated over a symmetric interval.
585 Easy
585 of 791
Application +4 | -1
The value of ∫1e2  dxx (1  +  log⁡  x)2\int \nolimits_1^{{e^2}} \,\,\frac{{dx}}{{x\,{{(1\,\, + \,\,\log \,\,x)}^2}}} is
View answer and explanation
Correct option: A
23\frac{2}{3}
Option A (23\frac{2}{3}) is correct because the definite integral evaluates to 2/3 after a simple substitution.

Explore Other Chapters in MATHEMATICS

View All →
SEQUENCE OF SERIES 1764 MCQs MATRICES 1363 MCQs INDEFINITE INTEGRATION 1179 MCQs PROBABILITY 1146 MCQs COMPLEX NUMBERS 980 MCQs DIFFERENTIAL EQUATIONS 977 MCQs
586 Easy
586 of 791
Application +4 | -1
Let f(x)=∫1x  2  −  t2 dtf\left( x \right) = \int \nolimits_1^x \,\,\sqrt {2\,\, - \,\,{t^2}} \,dt.〗 Then, real roots of the equation x2−f′(x)=0{x^2} - f'\left( x \right) = 0 are
View answer and explanation
Correct option: A
±  1\pm \,\,1
Option A (±  1\pm \,\,1) is correct because these are the real roots obtained by solving the equation x2−f′(x)=0{x^2} - f'\left( x \right) = 0 after finding f′(x)f'(x).
587 Easy
587 of 791
Application +4 | -1
The area bounded by the straight lines x=0,x=2x=0, x=2 and the curves y=2x,y=2x−x2y=2^x, y=2 x-x^2 is
View answer and explanation
Correct option: D
3log⁡2−43\frac{3}{\log 2}-\frac{4}{3}
Option D (3log⁡2−43\frac{3}{\log 2}-\frac{4}{3}) is correct because it represents the definite integral of the difference between the two functions between the specified limits.
588 Easy
588 of 791
Application +4 | -1
The value of ∫0πecos⁡xsin⁡x(1+cos⁡2x)(ecos⁡x+e−cos⁡x)dx\int_0^\pi \frac{e^{\cos x} \sin x}{\left(1+\cos ^2 x\right)\left(e^{\cos x}+e^{-\cos x}\right)} d x is equal to
View answer and explanation
Correct option: C
π4\frac{\pi}{4}
Option C (π4\frac{\pi}{4}) is correct because it is the result of evaluating the definite integral using the property of definite integrals and substitution.
589 Easy
589 of 791
Application +4 | -1
∫0π/2  cos⁡  x−  sin⁡  x1  + cos⁡  x  sin⁡  x  dx\int \nolimits_0^{\pi /2} \,\,\frac{{\cos \,\,x - \,\,\sin \,\,x}}{{1\,\, + \,\cos \,\,x\,\,\sin \,\,x}}\,\,dx is equal to
View answer and explanation
Correct option: A
0\mathrm{0}
Option A (0) is correct because using the property of definite integrals, the integral evaluates to zero.
590 Easy
590 of 791
Understanding +4 | -1
If Sn=e1/nn2+2n2e2/n+3n2e3/n+⋯+1neS_n=\frac{e^{1 / n}}{n^2}+\frac{2}{n^2} e^{2 / n}+\frac{3}{n^2} e^{3 / n}+\cdots+\frac{1}{n} e then lim⁡n→∞Sn\lim \limits_{{n \rightarrow \infty}} S_n is
View answer and explanation
Correct option: A
1
Option A (1) is correct because the limit of the given sum can be expressed as a definite integral which evaluates to 1.
591 Easy
591 of 791
Application +4 | -1
The value of ∫αβ x   ∣ x ∣  dx\int \limits_\alpha ^\beta \,x\,\;\left| {\,x\,} \right|\,\,dx, where α<0<β\alpha < 0 < \beta, is
View answer and explanation
Correct option: C
13  (α3+β3)\frac{1}{3}\,\,\left( {{\alpha ^3} + {\beta ^3}} \right)
Option D (None of these) is incorrect; the correct answer is 13  (α3+β3)\frac{1}{3}\,\,\left( {{\alpha ^3} + {\beta ^3}} \right), derived by splitting the integral and considering the absolute value.
592 Easy
592 of 791
Application +4 | -1
If y=f(x)y=f(x) be an invertible function with inverse gg and h(x)=xf(x)h(x)=x f(x), then ∫f(a)f(b)g(x)dx+∫abf(x)dx\int\limits_{{f(a)}}^{f(b)} g(x) d x+\int\limits_{a}^b f(x) d x is equal to
View answer and explanation
Correct option: B
h(b)−h(a)h(b)-h(a)
Option B (h(b)-h(a)) is correct because after substitution and integration by parts, the expression simplifies to this.
593 Easy
593 of 791
Analyzing +4 | -1
∫0π/2sin⁡2xlog⁡tan⁡x dx\int\limits_{0}^{\pi / 2} \sin 2 x \log \tan x \,d x is equal to
View answer and explanation
Correct option: C
0
Option C (0) is correct because the integral evaluates to zero due to the properties of the logarithm and trigonometric functions after applying a substitution.
594 Easy
594 of 791
Application +4 | -1
Area bounded between y=x⋅e−x2,y=0y=x \cdot e^{-x^2}, y=0 and max. ordinate is
View answer and explanation
Correct option: C
12(1−1e)\frac{1}{2}\left(1-\frac{1}{\sqrt{{e}}}\right)
Option C (\frac{{1}}{{2}}\left(1-\frac{{1}}{{\sqrt{{{e}}}}\right)\frac{{1}}{{2}}\left(1-\frac{{1}}{{\sqrt{{{e}}}}\right)) is correct because it is the area bounded between the curve and the x-axis up to the maximum ordinate.
595 Easy
595 of 791
Application +4 | -1
Let f(x)=x−[x]f\left( x \right) = x - \left[ x \right], for every real xx, where [xx]is the greatest integer less than or equal to xx. Then,∫−11  f  (x)dx  is\int \nolimits_{ - 1}^1 \,\,f\,\,\left( x \right)dx\,\,\mathrm{is}
View answer and explanation
Correct option: A
1\mathrm{1}
Option A (1\mathrm{1}) is correct because the integral of the fractional part of x over the interval [-1, 1] equals 1.
596 Easy
596 of 791
Application +4 | -1
Let ddx    F(x)=(esin⁡xx),  x>0\frac{d}{{dx}}\;\,\,F\left( x \right) = \left( {\frac{{{e^{\sin x}}}}{x}} \right),\,\,x > 0, If ∫14  3x  esin⁡x3  dx=F(k)  −F(1)\int \nolimits_1^4 \,\,\frac{3}{x}\,\,{e^{\sin {x^3}}}\;dx = F\left( k \right)\,\, - F\left( 1 \right), then one of the possible values of kk, is
View answer and explanation
Correct option: D
64\mathrm{64}
The correct answer is D (64) because after substitution and simplification, the value of k is found to be 64.
597 Easy
597 of 791
Analyzing +4 | -1
Let f(x)f(x) be an indefinite integral of cos⁡3x\cos ^3 x.

Statement 1:f(x)1: f(x) is a periodic function of period π\pi.

Statement 2: cos⁡3x\cos ^3 x is a periodic function.
View answer and explanation
Correct option: D
Statement 1 is false, Statement 2 is true.
Statement 1 is false, Statement 2 is true. cos3(x) is periodic, but its indefinite integral is not periodic with period pi.
598 Easy
598 of 791
Application +4 | -1
Let function FF be defined as F(x)=∫1xettdt,x>0F(x)=\int\limits_{1}^x \frac{\mathrm{e}^t}{\mathrm{t}} \mathrm{dt}, x > 0 then the value of the integral ∫1xett+adt\int\limits_{1}^x \frac{\mathrm{e}^t}{t+a} d t, where a>0a > 0, is:
View answer and explanation
Correct option: D
e−a[F(x+a)−F(1+a)]e^{-a}[F(x+a)-F(1+a)]
Option D (e−a[F(x+a)−F(1+a)]e^{-a}[F(x+a)-F(1+a)]) is correct because it is the result obtained after applying a suitable substitution and using the given definition of the function F(x).
599 Easy
599 of 791
Understanding +4 | -1
The value of the integral ∫−11  sin⁡11  x  dx\int \limits_{ - 1}^1 \,\,{\sin ^{11}}\,\,x\,\,dx, is
View answer and explanation
Correct option: D
0\mathrm{0}
Option D (0) is correct because sin⁡11(x)\sin^{11}(x) is an odd function, and the integral of an odd function over a symmetric interval is zero.
600 Easy
600 of 791
Analyzing +4 | -1
lim⁡n→∞(n!)1/nn\lim \limits_{{n \rightarrow \infty}} \frac{(n !)^{1 / n}}{n} equals
View answer and explanation
Correct option: B
e−1e^{-1}
Option A (12\frac{1}{2}) is correct because the limit can be expressed as a definite integral, which evaluates to 12\frac{1}{2}.
Revision notes and key takeaways

Review DEFINITE INTEGRATION in your textbook or class notes, then use this set to check your understanding.

Showing 571-600 of 791 questions

Continue Your Practice

📚 More Sets in This Chapter

Try different difficulty levels to master this topic

→ ENGINEERING MATHEMATICS - DEFINITE INTEGRATION MCQs (All) 995 MCQs → ENGINEERING MATHEMATICS - DEFINITE INTEGRATION MCQs (Hard) 51 MCQs → ENGINEERING MATHEMATICS - DEFINITE INTEGRATION MCQs (Medium) 596 MCQs
🎯 Next Chapters to Practice

Continue building your subject mastery

→ SEQUENCE OF SERIES 1764 MCQs → MATRICES 1363 MCQs → INDEFINITE INTEGRATION 1179 MCQs → PROBABILITY 1146 MCQs

Your practice

0 / 791 answered · 791 remaining · 0% accuracy

Progress is stored on this device. Sign in to save to your account.

✓ Correct · ✗ Incorrect · ○ Unanswered

Practice settings

Type at least 2 characters or jump directly:
🩺 NEET MCQs 🔬 JEE MCQs 🎯 JEE College Predictor Study guides 📊 Dashboard
🔓

Save your learning progress

Sign in with a free account to save your practice answers and quiz results, then return to them when you revise.

Continue with Free Account →