Skip to main content
Back to DEFINITE INTEGRATION

Your Progress

0 / 995 Answered
0
Correct
0
Incorrect
0%
Accuracy
JEE • MATHEMATICS • DEFINITE INTEGRATION

Definite integration practice · Mixed difficulty

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 721-750 of 995 questions

Preparing practice and saved progress…

721 Easy
721 of 995
Analyzing +4 | -1
If I1=∫03π  f(cos⁡2  x) dx{I_1} = \int \limits_0^{3\pi } \,\,f\left( {{{\cos }^2}\,\,x} \right)\,dx and I2=∫0π  f(cos⁡2  x)  dx{I_2} = \int \limits_0^\pi \,\,f\left( {{{\cos }^2}\,\,x} \right)\,\,dx then
View answer and explanation
Correct option: D
None of these
Option D (None of these) is correct because I1=3I2{I_1} = 3{I_2}, which is not represented by options A, B, or C.
722 Easy
722 of 995
Application +4 | -1
∫−22  (x  −  ∣ x ∣ )  dx\int \nolimits_{ - 2}^2 \,\,\left( {x\,\, - \,\,\left| {\,x\,} \right|\,} \right)\,\,dx is equal to
View answer and explanation
Correct option: D
− 4- \,4
Option D ( − 4\ - \,4) is correct because the integral evaluates to -4 after considering the definition of the absolute value function.
723 Easy
723 of 995
Application +4 | -1
If F(x)=∫3x(2+d dtcos⁡t)dtF(x)=\int\limits_{3}^x\left(2+\frac{d}{\mathrm{~d} t} \cos t\right) \mathrm{d} t then F′(π6)F^{\prime}\left(\frac{\pi}{6}\right) is equal to
View answer and explanation
Correct option: D
3/23 / 2
Option D (3/2) is correct because the derivative of the integral is found using the Fundamental Theorem of Calculus and evaluating at x = π/6.
724 Easy
724 of 995
Application +4 | -1
The value of ∫14  ∣ x−3 ∣dx\int \nolimits_1^4 \,\,\left| {\,x - 3\,} \right|dx is equal to
View answer and explanation
Correct option: B
52\frac{5}{2}
Option B is correct because the absolute value function requires splitting the integral at x=3, and then integrating each part separately, resulting in 5/2.
725 Easy
725 of 995
Application +4 | -1
∫0π/4log⁡(1+tan⁡2θ+2tan⁡θ)dθ=\int\limits_{0}^{\pi / 4} \log \left(1+\tan ^2 \theta+2 \tan \theta\right) d \theta=
View answer and explanation
Correct option: C
(πlog⁡2)/4(\pi \log 2) / 4
Option C is correct because the integral is solved using properties of logarithms and definite integrals, specifically using substitution and the property of integrals with limits 0 to a.
726 Medium
726 of 995
Analyzing +4 | -1
Let R→R\mathbb{R} \rightarrow \mathbb{R} be function defined as f(x)=a sin⁡(π[x]2)+[2−x],a∈Rf(x)=\operatorname{a\,sin}\left(\frac{\pi[x]}{2}\right)+[2-x], a \in \mathbb{R}, where [t][t] is the greatest integer less than or equal to tt. If lim⁡x→−1f(x)\lim _{x \rightarrow-1} f(x) exists, then the value of ∫04f(x)dx\int_0^4 f(x) d x is equal to
View answer and explanation
Correct option: B
−2-2
Option D (2) is correct because by finding the value of 'a' using the limit condition and then integrating the function from 0 to 4, we get the result as -2.
727 Easy
727 of 995
Application +4 | -1
The value of ∫01 x31+x8  dx\int \limits_0^1 \,\frac{{{x^3}}}{{1 + {x^8}}}\,\,dx, is
View answer and explanation
Correct option: C
π16\frac{\pi }{16}
Option C (π16\frac{\pi }{16}) is correct because it is the result of substituting x4=tx^4 = t and evaluating the resulting integral.
728 Easy
728 of 995
Application +4 | -1
The value of I=∫−π/2π/2cos⁡x−cos⁡3x dxI=\int\limits_{{-\pi / 2}}^{\pi / 2} \sqrt{\cos x-\cos ^3 x} \mathrm{~d} x is
View answer and explanation
Correct option: C
4/34 / 3
Option C (4/3) is correct because the integral simplifies to 2∫0π/2cos⁡xsin⁡xdx2 \int_0^{\pi/2} \sqrt{{\cos x}} \sin x dx, which can be easily evaluated using substitution.
729 Easy
729 of 995
Application +4 | -1
Let g(x)={1,0≤x<1x3,1≤x<4x,4≤x<9g(x)=\left\{\begin{array}{cl}1, & 0 \leq x < 1 \\ x^3, & 1 \leq x < 4 \\ \sqrt{x}, & 4 \leq x < 9\end{array}\right. then ∫09g(x)dx\int\limits_{0}^9 g(x) \mathrm{d} x is
View answer and explanation
Correct option: C
929/12929 / 12
Option C is correct because the integral can be solved by splitting the integral based on the piecewise function definition.
730 Easy
730 of 995
Application +4 | -1
∫0π/2  sin⁡2  xsin⁡  x  +  cos⁡  x  dx\int \limits_0^{\pi /2} \,\,\frac{{{{\sin }^2}\,\,x}}{{\sin \,\,x\,\, + \,\,\cos \,\,x}}\,\,dx is equal to
View answer and explanation
Correct option: C
12  log⁡  (2+1)\frac{1}{{\sqrt 2 }}\,\,\log \,\,\left( {\sqrt 2 + 1} \right)
Option C (12  log⁡  (2+1)\frac{1}{{\sqrt 2 }}\,\,\log \,\,\left( {\sqrt 2 + 1} \right)) is correct because using the property of definite integrals and simplifying leads to the given answer.
731 Easy
731 of 995
Application +4 | -1
For x>0x > 0, let f(x)=∫1xlog⁡t1+tdtf(x)=\int\limits_{1}^x \frac{\log t}{1+\mathrm{t}} \mathrm{dt}. Then f(x)+f(1x)f(x)+f\left(\frac{1}{x}\right) is equal to
View answer and explanation
Correct option: C
12(log⁡x)2\frac{1}{2}(\log x)^2
Option C (12(log⁡x)2\frac{1}{2}(\log x)^2) is correct because it is the result obtained after simplifying the expression f(x)+f(1x)f(x) + f(\frac{1}{x}) using a suitable substitution and evaluating the resulting integral.
732 Easy
732 of 995
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.
733 Easy
733 of 995
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.
734 Medium
734 of 995
Application +4 | -1
∫π/43π/4dx1+cos⁡x\int_{\pi /4}^{3\pi /4} {\frac{{dx}}{{1 + \cos x}}} is equal to
View answer and explanation
Correct option: B
2
Option C (1.1×10−9M1.1 \times {10^{ - 9}}M) is correct because it accounts for the dilution effect when calculating the original concentration of Ba2+Ba^{2+}.
735 Medium
735 of 995
Application +4 | -1
The value of the definite integral ∫a+2πa+5π/2  (sin⁡−1  (cos⁡  x)+cos⁡−1  (sin⁡ x))  dx\int \nolimits_{a + 2\pi }^{a + 5\pi /2} \,\,({\sin ^{ - 1}}\,\,(\cos \,\,x) + {\cos ^{ - 1}}\,\,(\sin \,x))\;dx is equal to
View answer and explanation
Correct option: B
π24\frac{{{\mathrm{\pi }^2}}}{4}
Option B (π24\frac{{{\mathrm{\pi }^2}}}{4}) is correct because after simplifying the integrand using inverse trigonometric identities and evaluating the integral, the result is this value.

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
736 Easy
736 of 995
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.
737 Medium
737 of 995
Application +4 | -1
∫02π11+tan⁡4xdx\int_0^{2 \pi} \frac{1}{1+\tan ^4 x} d x is equal to
View answer and explanation
Correct option: A
π\pi
Option A (π) is correct because using properties of definite integrals and symmetry, the integral simplifies to π.
738 Hard
738 of 995
Analyzing +4 | -1
Consider the integrals
I1=∫01 e−xcos⁡2  x dx,    I2=∫01 e−x2cos⁡2  x dx,    I3=∫01 e−x2  dx{I_1} = \int \limits_0^1 \,{e^{ - x}}{\cos ^2}\,\,x\,dx,\;\,\,{I_2} = \int \limits_0^1 \,{e^{ - {x^2}}}{\cos ^2}\,\,x\,dx,\;\,\,{I_3} = \int \limits_0^1 \,{e^{ - {x^2}}}\;dx and I4=∫01 e−(1/2)x2  dx{I_4} = \int \limits_0^1 \,{e^{ - \left( {1/2} \right){x^2}}}\,\,dx The greatest of these integral is
View answer and explanation
Correct option: D
I4{I_4}
Option D (I4{I_4}) is correct because by comparing the integrands over the interval [0,1], e−(1/2)x2{e^{ - (1/2){x^2}}} is the greatest, leading to the largest integral value.
739 Easy
739 of 995
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.
740 Easy
740 of 995
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.
741 Easy
741 of 995
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}}.
742 Easy
742 of 995
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.
743 Easy
743 of 995
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.
744 Easy
744 of 995
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.
745 Medium
745 of 995
Application +4 | -1
The value of the integral ∫−π/2π/2{x2+log⁡(π+xπ−x)}cos⁡x dx, is\int_{ - \pi /2}^{\pi /2} {\left\{ {{x^2} + \log \left( {\frac{{\pi + x}}{{\pi - x}}} \right)} \right\}} \cos x\,dx,\,\mathrm{ is }
View answer and explanation
Correct option: B
π22−4\frac{\pi^2}{2}-4
The correct answer is B (π22−4\frac{\pi^2}{2}-4) because after splitting the integral and evaluating, the result is π22−4\frac{\pi^2}{2}-4.
746 Easy
746 of 995
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.
747 Easy
747 of 995
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.
748 Easy
748 of 995
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.
749 Easy
749 of 995
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).
750 Medium
750 of 995
Application +4 | -1
∫0π  x  sin⁡4  x dx\int \nolimits_0^\pi \,\,x\,\,{\sin ^4}\,\,x\,dx is equal to
View answer and explanation
Correct option: B
3 π216\frac{{3\,{\pi ^2}}}{{16}}
Option B (3 π216\frac{{3\,{\pi ^2}}}{{16}}) is correct because the integral is solved using properties of definite integrals and trigonometric identities.
Revision notes and key takeaways

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

Showing 721-750 of 995 questions

Continue Your Practice

📚 More Sets in This Chapter

Try different difficulty levels to master this topic

→ ENGINEERING MATHEMATICS - DEFINITE INTEGRATION MCQs (Easy) 791 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 / 995 answered · 995 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 →