JEE Main Maths · Integrals

Integration by Substitution in JEE Main

Name: Integration by Substitution: JEE Main previous year questions
Creator: Lumi

Integration by Substitution (Maths, Integrals) came 22 times in 20 of the 42 JEE Main shifts Lumi analysed, across 15 distinct ideas. 12 of them came in 2024–26; it was last asked in 2026.

Questions: 22; in 20 shifts
Ideas asked: 15; 4 asked more than once
Numerical answer: 7; 32% 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 1Medium 7Hard 14

Format

Single correct15 Q
Numerical value7 Q

Ideas inside the topic

15 ideas

Composite Function Substitution5 Q
Exponential Factor Manipulation2 Q
Factoring Highest Power for Substitution2 Q
Even Power of Secant Substitution2 Q
Dividing by Cosine Squared Substitution1 Q
Even Secant/Cosecant Power Substitution1 Q
Exact Exponential-Function Derivative Substitution1 Q
Fractional Powers LCM Substitution1 Q
Hyperbolic Substitution1 Q
Numerator-Denominator Derivative Matching1 Q
Odd Power of Cosine Substitution1 Q
Odd Power of Cosine Substitution1 Q

Where it sits

Part of Integrals, which had 73 questions in all. Integration by Substitution accounts for 30% of them. In Lumi's JEE taxonomy it belongs to Calculus.

Some questions sit in other chapters too: Differential Equations.

Every question

All 22 Integration by Substitution questions

Newest first, each with its options and official answer.

5 Apr 2026, Shift 1 · Q20The value of the integral $\int_{\pi/6}^{\pi/3}\left(\frac{4 - \operatorname{cosec}^2 x}{\cos^4 x}\right)dx$ is:MediumSingle correct
5 Apr 2026, Shift 2 · Q18Let $(2^{1-a}+2^{1+a})$ , $f(a)$ , $(3^a+3^{-a})$ be in A.P. and $\alpha$ be the minimum value of $f(a)$ . Then the value of the integral…HardSingle correct
3 Apr 2025, Shift 1 · Q18Let $f(x) = \int x^3\sqrt{3 - x^2}\,dx$ . If $5f(\sqrt{2}) = -4$ , then $f(1)$ is equal toMediumSingle correct
4 Apr 2025, Shift 2 · Q25If…HardNumerical value
7 Apr 2025, Shift 2 · Q24If…HardNumerical value
6 Apr 2024, Shift 2 · Q12If $\int\frac{1}{a^2\sin^2x+b^2\cos^2x}dx=\frac{1}{12}\tan^{-1}(3\tan x)+\text{constant}$ , then the maximum value of $a\sin x+b\cos x$ , is :MediumSingle correct
8 Apr 2024, Shift 1 · Q10Let $I(x)=\int\frac{6}{\sin^2x(1-\cot x)^2}dx$ . If $I(0)=3$ , then $I\left(\frac{\pi}{12}\right)$ is equal toMediumSingle correct
9 Apr 2024, Shift 1 · Q8Let $\int\frac{2-\tan x}{3+\tan x}\,dx=\frac{1}{2}\left(\alpha x+\log_e|\beta\sin x+\gamma\cos x|\right)+C$ , where C is the constant of…MediumSingle correct
29 Jan 2024, Shift 1 · Q12For $x\in\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$ , if…HardSingle correct
30 Jan 2024, Shift 2 · Q12Let $y = f(x)$ be a thrice differentiable function in $(-5, 5)$ . Let the tangents to the curve $y = f(x)$ at $(1, f(1))$ and $(3, f(3))$ …HardSingle correct
30 Jan 2024, Shift 2 · Q13Let $f: \mathbb{R} \to \mathbb{R}$ be a function defined by $f(x) = \frac{x}{(1+x^4)^{1/4}}$ , and $g(x) = f(f(f(f(x))))$ . Then,…HardSingle correct
31 Jan 2024, Shift 1 · Q27If the integral $525\int_0^{\frac{\pi}{2}}\sin2x\cos^{\frac{11}{2}}x\left(1+\cos^{\frac{5}{2}}x\right)^{\frac{1}{2}}dx$ is equal to…HardNumerical value
13 Apr 2023, Shift 2 · Q26Let $f_n=\int_0^{\frac{\pi}{2}}\left(\sum_{k=1}^{n}\sin^{k-1}x\right)\left(\sum_{k=1}^{n}(2k-1)\sin^{k-1}x\right)\cos x\,dx$ ,…HardNumerical value
25 Jul 2022, Shift 1 · Q10The slope of the tangent to a curve $C:y=y(x)$ at any point $(x,y)$ on it is…HardSingle correct
28 Jul 2022, Shift 1 · Q24If $\int_0^{\sqrt{3}}\frac{15x^3}{\sqrt{1+x^2+\sqrt{(1+x^2)^3}}}\,dx=\alpha\sqrt{2}+\beta\sqrt{3}$ , where $\alpha,\beta$ are integers,…HardNumerical value
16 Mar 2021, Shift 2 · Q81For real numbers $\alpha,\beta,\gamma$ and $\delta$ , if…HardNumerical value
18 Mar 2021, Shift 1 · Q73The integral $\int\dfrac{(2x-1)\cos\sqrt{(2x-1)^2+5}}{\sqrt{4x^2-4x+6}}\,dx$ is equal to : (where c is a constant of integration)MediumSingle correct
18 Mar 2021, Shift 1 · Q83If $f(x)=\int\dfrac{5x^8+7x^6}{(x^2+1+2x^7)^2}\,dx$ , $(x\ge0)$ , $f(0)=0$ and $f(1)=\dfrac{1}{K}$ , then the value of K is __________.HardNumerical value
8 Jan 2020, Shift 1 · Q60If $\int\dfrac{\cos x\,dx}{\sin^3 x\,(1+\sin^6 x)^{2/3}}=f(x)\,(1+\sin^6 x)^{1/\lambda}+c$ where c is a constant of integration, then…HardSingle correct
9 Jan 2020, Shift 2 · Q62If $\int\frac{d\theta}{\cos^2\theta(\tan2\theta+\sec2\theta)}=\lambda\tan\theta+2\log_e|f(\theta)|+C$ where $C$ is a constant of…MediumSingle correct
6 Sep 2020, Shift 2 · Q61The integral $\int_1^2 e^x\cdot x^x(2+\log_e x)\,dx$ equals :HardSingle correct
2 Apr 2017 (offline) · Q75The integral $\int_{\pi/4}^{3\pi/4}\frac{dx}{1+\cos x}$ is equal to :EasySingle correct