Definite Integration

24 JEE Main questions · Maths

MathsJEE Main 21 Jan 2026, Shift 1·Q12·MCQ·Moderate
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Definite integral properties
The value of $\displaystyle\int_{-\pi/6}^{\pi/6}\left(\frac{\pi + 4x^{11}}{1 - \sin(\lvert x\rvert + \pi/6)}\right)dx$ is equal to
1. (1) $2\pi$
2. (2) $4\pi$
3. (3) $8\pi$
4. (4) $6\pi$
MathsJEE Main 21 Jan 2026, Shift 1·Q25·Numerical·Moderate
Open
Integral of modulus function
$6\displaystyle\int_0^{\pi}\left\lvert(\sin 3x + \sin 2x + \sin x)\right\rvert dx$ is equal to .
MathsJEE Main 21 Jan 2026, Shift 2·Q22·Numerical·Tough
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Definite integral of inverse trigonometric functions
If $\displaystyle\int_0^1 4\cot^{-1}(1 - 2x + 4x^2)\,dx = a\tan^{-1}(2) - b\log_e(5)$ , where $a, b \in \mathbb{N}$ , then $(2a + b)$ is equal to .
MathsJEE Main 22 Jan 2026, Shift 1·Q15·MCQ·Easy–moderate
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Greatest integer function integral
The value of $\displaystyle\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\left(\frac{1}{[x] + 4}\right)dx$ , where $[\bullet]$ denotes the greatest integer function, is
1. (1) $\dfrac{1}{60}(21\pi - 1)$
2. (2) $\dfrac{1}{60}(\pi - 7)$
3. (3) $\dfrac{7}{60}(3\pi - 1)$
4. (4) $\dfrac{7}{60}(\pi - 3)$
MathsJEE Main 22 Jan 2026, Shift 2·Q22·Numerical·Tough
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Definite integral properties
Let $[\,\bullet\,]$ be the greatest integer function. If $\alpha = \displaystyle\int_0^{64}\left(x^{1/3} - \left[x^{1/3}\right]\right)dx$ , then $\frac{1}{\pi}\int_0^{\alpha\pi}\left(\frac{\sin^2\theta}{\sin^6\theta + \cos^6\theta}\right)d\theta$ is equal to .
MathsJEE Main 23 Jan 2026, Shift 1·Q10·MCQ·Easy–moderate
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Definite integral properties
The value of the integral $\displaystyle\int_{\frac{\pi}{24}}^{\frac{5\pi}{24}}\frac{dx}{1 + \sqrt[3]{\tan 2x}}$ is:
1. (1) $\dfrac{\pi}{12}$
2. (2) $\dfrac{\pi}{18}$
3. (3) $\dfrac{\pi}{6}$
4. (4) $\dfrac{\pi}{3}$
MathsJEE Main 23 Jan 2026, Shift 2·Q23·Numerical·Moderate
Open
Integral with variable limit
The number of elements in the set $S = \left\{x : x \in [0, 100] \text{ and } \int_0^x t^2\sin(x - t)\,dt = x^2\right\}$ is .
MathsJEE Main 24 Jan 2026, Shift 2·Q21·Numerical·Moderate
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Integral equations with definite integrals
If $f(x)$ satisfies the relation $f(x) = e^x + \displaystyle\int_0^1\left(y + xe^x\right)f(y)\,dy$ , then $e + f(0)$ is equal to .
MathsJEE Main 28 Jan 2026, Shift 1·Q12·MCQ·Easy–moderate
Open
Definite integral of polynomial
Let $f$ be a polynomial function such that $f(x^2 + 1) = x^4 + 5x^2 + 2$ , for all $x \in \mathbb{R}$ . Then $\displaystyle\int_0^3 f(x)\,dx$ is equal to
1. (1) $\dfrac{41}{3}$
2. (2) $\dfrac{33}{2}$
3. (3) $\dfrac{27}{2}$
4. (4) $\dfrac{5}{3}$
MathsJEE Main 28 Jan 2026, Shift 1·Q24·Numerical·Moderate
Open
Definite integral properties
The value of $\displaystyle\sum_{r=1}^{20}\left(\left\lvert\sqrt{\pi\left(\int_0^r x\,\lvert\sin\pi x\rvert\,dx\right)}\right\rvert\right)$ is .
MathsJEE Main 28 Jan 2026, Shift 2·Q13·MCQ·Moderate
Open
Greatest integer function integral
Let $[\cdot]$ denote the greatest integer function. Then $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\left(\dfrac{12(3 + [x])}{3 + [\sin x] + [\cos x]}\right)dx$ is equal to:
1. (1) $15\pi + 4$
2. (2) $11\pi + 2$
3. (3) $13\pi + 1$
4. (4) $12\pi + 5$
MathsJEE Main 2 Apr 2026, Shift 1·Q19·MCQ·Easy–moderate
Open
Integral with greatest integer function
Let $[\cdot]$ denote the greatest integer function. Then the value of $\displaystyle\int_0^3\left(\frac{e^x + e^{-x}}{[x]!}\right)dx$ is:
1. (1) $e^2 + e^3 - \dfrac{1}{e^2} - \dfrac{1}{e^3}$
2. (2) $\dfrac12\left(e^2 + e^3 - \dfrac{1}{e^2} - \dfrac{1}{e^3}\right)$
3. (3) $e^2 + e^3 - \dfrac{1}{2e^2} - \dfrac{1}{2e^3}$
4. (4) $\dfrac12(e^2 + e^3) - \dfrac{1}{e^2} - \dfrac{1}{e^3}$
MathsJEE Main 2 Apr 2026, Shift 1·Q25·Numerical·Moderate
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Integral of inverse function
If $\alpha = \displaystyle\int_0^{2\sqrt3}\log_2\left(x^2 + 4\right)dx + \int_2^4\sqrt{2^x - 4}\,dx$ , then $\alpha^2$ is equal to .
MathsJEE Main 2 Apr 2026, Shift 2·Q17·MCQ·Easy–moderate
Open
Integral of periodic function
The value of $\displaystyle\int_0^{20\pi}(\sin^4x + \cos^4x)\,dx$ is equal to:
1. (1) $\dfrac{15\pi}{2}$
2. (2) $25\pi$
3. (3) $15\pi$
4. (4) $\dfrac{25\pi}{2}$
MathsJEE Main 4 Apr 2026, Shift 1·Q19·MCQ·Tough
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Integral with greatest integer function
Let $\displaystyle\int_{-2}^{2}\left(\lvert\sin x\rvert + [x\sin x]\right)dx = 2(3 - \cos 2) + \beta$ , where $[\cdot]$ is the greatest integer function. Then $\beta\sin\left(\dfrac{\beta}{2}\right)$ equals.
1. (1)1
2. (2)2
3. (3)4
4. (4)8
MathsJEE Main 4 Apr 2026, Shift 2·Q20·MCQ·Moderate
Open
Inverse trigonometric integrals
The integral $\displaystyle\int_0^1\cot^{-1}\left(1 + x + x^2\right)dx$ is equal to:
1. (1) $2\tan^{-1}2 + \dfrac12\log_e\left(\dfrac54\right) + \dfrac{\pi}{2}$
2. (2) $2\tan^{-1}2 + \dfrac12\log_e\left(\dfrac54\right) - \dfrac{\pi}{2}$
3. (3) $2\tan^{-1}2 - \dfrac12\log_e\left(\dfrac54\right) + \dfrac{\pi}{2}$
4. (4) $2\tan^{-1}2 - \dfrac12\log_e\left(\dfrac54\right) - \dfrac{\pi}{2}$
MathsJEE Main 5 Apr 2026, Shift 1·Q18·MCQ·Moderate
Open
Definite integral by substitution
The value of the integral $\displaystyle\int_0^{\infty}\frac{\log_e(x)}{x^2 + 4}\,dx$ is:
1. (1) $\dfrac{\pi\log_e(2)}{2}$
2. (2) $\dfrac{\pi\log_e(2)}{4}$
3. (3) $1 + \pi\log_e(2)$
4. (4) $2 + \pi\log_e(2)$
MathsJEE Main 5 Apr 2026, Shift 1·Q20·MCQ·Moderate
Open
Definite integral evaluation
The value of the integral $\displaystyle\int_{\pi/6}^{\pi/3}\left(\frac{4 - \operatorname{cosec}^2x}{\cos^4x}\right)dx$ is:
1. (1) $\dfrac{11}{\sqrt3}$
2. (2) $\dfrac{16}{\sqrt3}$
3. (3) $\dfrac{32}{3\sqrt3}$
4. (4) $\dfrac{64}{3\sqrt3}$
MathsJEE Main 5 Apr 2026, Shift 2·Q18·MCQ·Moderate
Open
Definite integral by substitution
Let $\left(2^{1-a} + 2^{1+a}\right)$ , $f(a)$ , $\left(3^{a} + 3^{-a}\right)$ be a A.P. and $\alpha$ be the minimum value of $f(a)$ . Then the value of the integral $\int_{\log_e(\alpha-1)}^{\log_e(\alpha)} \frac{dx}{\left(e^{2x} - e^{-2x}\right)}$ is:
1. (1) $\dfrac12\log_e\left(\dfrac43\right)$
2. (2) $\dfrac14\log_e\left(\dfrac43\right)$
3. (3) $\dfrac12\log_e\left(\dfrac85\right)$
4. (4) $\dfrac14\log_e\left(\dfrac85\right)$
MathsJEE Main 6 Apr 2026, Shift 1·Q18·MCQ·Moderate
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Definite integral properties
The value of the integral $\displaystyle\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}}\left(\frac{32\cos^4x}{1 + e^{\sin x}}\right)dx$ is:
1. (1) $4\pi + 2$
2. (2) $3\pi + 8$
3. (3) $3\pi + 4$
4. (4) $4\pi + 3$
MathsJEE Main 6 Apr 2026, Shift 2·Q20·MCQ·Easy–moderate
Open
Definite integral properties
The value of the integral $\displaystyle\int_{-1}^{1}\left(\frac{x^3 + \lvert x\rvert + 1}{x^2 + 2\lvert x\rvert + 1}\right)dx$ is equal to
1. (1) $3\log_e2$
2. (2) $2\log_e2$
3. (3) $5\log_e3$
4. (4) $3\log_e3$
MathsJEE Main 8 Apr 2026, Shift 2·Q16·MCQ·Moderate
Open
Definite integral by substitution
The value of the integral $\int_0^2 \frac{\sqrt{x(x^2 + x + 1)}}{\left(\sqrt{x+1}\right)\left(\sqrt{x^4 + x^2 + 1}\right)}\,dx$ is equal to:
1. (1) $\dfrac13\log_e\left(3 - 2\sqrt2\right)$
2. (2) $\dfrac23\log_e\left(4 + \sqrt2\right)$
3. (3) $\dfrac23\log_e\left(3 + 2\sqrt2\right)$
4. (4) $\dfrac13\log_e\left(1 + 6\sqrt2\right)$
MathsJEE Main 8 Apr 2026, Shift 2·Q19·MCQ·Moderate
Open
Integral of modulus function
Let $f(x) = \begin{cases} \dfrac13, & x \le \dfrac{\pi}{2}\\[2ex] \dfrac{b(1 - \sin x)}{(\pi - 2x)^2}, & x > \dfrac{\pi}{2}\end{cases},$ If $f$ is continuous at $x = \pi/2$ , then the value of $\displaystyle\int_0^{3b-6}\left\lvert x^2 + 2x - 3\right\rvert dx$ is:
1. (1)5
2. (2)2
3. (3)3
4. (4)4
MathsJEE Main 8 Apr 2026, Shift 2·Q22·Numerical·Moderate
Open
Trigonometric definite integrals
If $\int_{\pi/6}^{\pi/4}\left(\cot\left(x - \frac{\pi}{3}\right)\cot\left(x + \frac{\pi}{3}\right) + 1\right)dx = \alpha\log_e\left(\sqrt3 - 1\right),$ then $9\alpha^2$ is equal to .