JEE Main Maths · Integrals

Definite Integral Properties in JEE Main

Name: Definite Integral Properties: JEE Main previous year questions
Creator: Lumi

Definite Integral Properties (Maths, Integrals) came 30 times in 27 of the 42 JEE Main shifts Lumi analysed, across 12 distinct ideas. 20 of them came in 2024–26; it was last asked in 2026.

Questions: 30; in 27 shifts
Ideas asked: 12; 8 asked more than once
Numerical answer: 10; 33% of its questions
Pattern: Asked most years; last asked 2026

Questions each year

In the shifts Lumi analysed for each year.

Difficulty and format

Easy 2Medium 13Hard 15

Format

Single correct20 Q
Numerical value10 Q

Ideas inside the topic

12 ideas

Greatest Integer Function Integrals9 Q
Even and Odd Functions3 Q
King's Property3 Q
Symmetric Limits with Exponential Denominator Trick3 Q
Modulus Function Interval Splitting2 Q
Periodic Function Property2 Q
Reduction Formulae in Definite Integrals2 Q
Removing 'x' using King's Property2 Q
Integral of Inverse Functions1 Q
Reciprocal Substitution (x = 1/t)1 Q
King's Property Rational Shortcut1 Q
Max/Min Function Integrals1 Q

Where it sits

Part of Integrals, which had 73 questions in all. Definite Integral Properties accounts for 41% of them. In Lumi's JEE taxonomy it belongs to Calculus.

Every question

All 30 Definite Integral Properties questions

Newest first, each with its options and official answer.

2 Apr 2026, Shift 1 · Q19Let $[\cdot]$ denote the greatest integer function. Then the value of $\int_0^3\left(\frac{e^x+e^{-x}}{[x]!}\right)dx$ is :MediumSingle correct
2 Apr 2026, Shift 1 · Q25If $\alpha=\int_0^{2\sqrt3}\log_2\left(x^2+4\right)dx+\int_2^4\sqrt{2^x-4}\,dx$ , then $\alpha^2$ is equal to __________.HardNumerical value
2 Apr 2026, Shift 2 · Q17The value of $\int_0^{20\pi} (\sin^4 x + \cos^4 x)dx$ is equal to:EasySingle correct
4 Apr 2026, Shift 1 · Q19Let $\int_{-2}^{2}\left(|\sin x|+[x\sin x]\right)dx=2(3-\cos 2)+\beta$ , where $[\cdot]$ is the greatest integer function. Then…HardSingle correct
5 Apr 2026, Shift 1 · Q18The value of the integral $\int_0^{\infty} \frac{\log_e(x)}{x^2+4}\,dx$ is:HardSingle correct
6 Apr 2026, Shift 1 · Q18The value of the integral $\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}}\left(\frac{32\cos^4 x}{1+e^{\sin x}}\right)dx$ is:MediumSingle correct
6 Apr 2026, Shift 2 · Q20The value of the integral $\int_{-1}^{1}\left(\frac{x^3+|x|+1}{x^2+2|x|+1}\right)dx$ is equal to :MediumSingle correct
2 Apr 2025, Shift 1 · Q23Let $[\cdot]$ denote the greatest integer function. If $\int_0^{e^3}\left[\frac{1}{e^{x-1}}\right]dx=\alpha-\log_e2$ , then $\alpha^3$ is…MediumNumerical value
2 Apr 2025, Shift 2 · Q19Let $(a, b)$ be the point of intersection of the curve $x^2 = 2y$ and the straight line $y - 2x - 6 = 0$ in the second quadrant. Then the…MediumSingle correct
3 Apr 2025, Shift 1 · Q19Let the domain of the function $f(x) = \log_2 \log_4 \log_6 (3 + 4x - x^2)$ be $(a, b)$ . If…HardSingle correct
3 Apr 2025, Shift 2 · Q18The integral $\int_0^{\pi} \frac{8x\,dx}{4\cos^2 x + \sin^2 x}$ is equal toMediumSingle correct
7 Apr 2025, Shift 1 · Q18The integral $\int_0^{\pi} \frac{(x+3)\sin x}{1+3\cos^2 x}\,dx$ is equal toMediumSingle correct
8 Apr 2025, Shift 2 · Q18Let $f(x)$ be a positive function and $I_1 = \int_{-\frac{1}{2}}^{1} 2x\, f\big(2x(1-2x)\big)\,dx$ and…HardSingle correct
8 Apr 2025, Shift 2 · Q19The integral $\int_{-1}^{\frac{3}{2}} \left(\left|\pi^2 x \sin(\pi x)\right|\right) dx$ is equal to :MediumSingle correct
6 Apr 2024, Shift 2 · Q25Let $[t]$ denote the largest integer less than or equal to $t$ . If…HardNumerical value
8 Apr 2024, Shift 1 · Q11The value of $k\in\mathbb{N}$ for which the integral $I_n=\int_0^1(1-x^k)^n dx$ , $n\in\mathbb{N}$ , satisfies $147I_{20}=148I_{21}$ isHardSingle correct
29 Jan 2024, Shift 1 · Q11If the value of the integral…HardSingle correct
31 Jan 2024, Shift 1 · Q26Let $f:\mathbb{R}\to\mathbb{R}$ be a function defined by $f(x)=\frac{4^x}{4^x+2}$ and…HardNumerical value
31 Jan 2024, Shift 2 · Q12Let $f,g:(0,\infty)\to\mathbb{R}$ be two functions defined by $f(x)=\int_{-x}^{x}\left(|t|-t^2\right)e^{-t^2}dt$ and…HardSingle correct
31 Jan 2024, Shift 2 · Q26 $\left|\frac{120}{\pi^3}\int_0^{\pi}\frac{x^2\sin x\cos x}{\sin^4x+\cos^4x}dx\right|$ is equal to ____________.HardNumerical value
8 Apr 2023, Shift 2 · Q26Let $[t]$ denote the greatest integer function. If $\int_0^{2.4}[x^2]\,dx = \alpha + \beta\sqrt{2} + \gamma\sqrt{3} + \delta\sqrt{5}$ ,…MediumNumerical value
13 Apr 2023, Shift 2 · Q10The value of…HardSingle correct
25 Jul 2022, Shift 1 · Q9For any real number $x$ , let $[x]$ denote the largest integer less than equal to $x$ . Let $f$ be a real valued function defined on the…HardSingle correct
24 Jun 2022, Shift 1 · Q27Let $\max_{0\le x\le 2}\left\{\frac{9-x^2}{5-x}\right\}=\alpha$ and $\min_{0\le x\le 2}\left\{\frac{9-x^2}{5-x}\right\}=\beta$ . If…HardNumerical value
29 Jun 2022, Shift 1 · Q17 $\int_0^5 \cos\left(\pi\left(x - \left[\frac{x}{2}\right]\right)\right)dx$ , where $[t]$ denotes greatest integer less than or equal to…MediumSingle correct
3 Aug 2021, Shift 2 · Q84If $[x]$ denotes the greatest integer $\le x$ , then the value of $\int_{-2}^{4}\left|[x] - x\right|dx$ is $\underline{\hspace{2em}}$ .MediumNumerical value
25 Jul 2021, Shift 1 · Q78The value of the definite integral $\int_{\pi/24}^{5\pi/24}\frac{dx}{1+\sqrt[3]{\tan2x}}$ is :MediumSingle correct
16 Mar 2021, Shift 2 · Q74Consider the integral $I=\int_0^{10}\frac{[x]\,e^{[x]}}{e^{x-1}}\,dx$ , where $[x]$ denotes the greatest integer less than or equal to $x$ .…MediumSingle correct
18 Mar 2021, Shift 1 · Q82Let $f(x)$ and $g(x)$ be two functions satisfying $f(x^2)+g(4-x)=4x^3$ and $g(4-x)+g(x)=0$ , then the value of $\int_{-4}^{4}f(x^2)\,dx$ is…HardNumerical value
2 Sep 2020, Shift 1 · Q73The integral $\int_{0}^{2}\left||x-1|-x\right|dx$ is equal to ___________.EasyNumerical value