JEE Main · Maths · Integrals

Integration by Substitution: JEE Main previous year questions

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

51 questions in the 108 JEE Main shifts Lumi analysed, in 8 of 8 years. Pattern: Asked most years. 16 came in 2024–26. Here is how it is asked and which ideas come back.

Questions: 51; in 46 of 108 shifts
Years asked: 8/8; last asked 2026
Chance in a shift: 47%; calibrated model, how we compute it
Weighted estimate: 0.4; model; history: 0.5 a shift on average, typically 0.2–0.6

Key findings

01
Asked in 8 of 8 years with analysed shifts and in 46 of 108 shifts: 51 questions, #53 of 196 topics. It came in every year with analysed shifts.
02
58% of its questions since 2020 tested an idea no earlier year had asked; 9 repeated an idea from the year before.
03
The most repeated idea is Composite Function Substitution, asked 6 times in 5 shifts (2021, 2024, 2025), most often as “indefinite integral with radical then evaluate constant”.
04
JEE Main has asked 27 of the 66 ideas in this topic; 34 have never been asked, directly or inside another question.
05
Mostly asked as single correct (69%); 31% need a numerical answer, 4% are easy and 61% hard.
06
The skill tested most is multi-step (71%), then calculation (27%).

Questions a shift, year by year

Out of the 25 Maths questions in a shift. Years differ in how many shifts Lumi analysed, so each year is scaled to one shift.

Raw count: 2017: 1 in 1 shift · 2020: 7 in 12 shifts · 2021: 14 in 24 shifts · 2022: 6 in 24 shifts · 2023: 7 in 12 shifts · 2024: 11 in 18 shifts · 2025: 3 in 8 shifts · 2026: 2 in 9 shifts.

Up to 2023: 0.5; a shift, of 25
Since 2024: 0.4; -0.2
Usual gap: 1.3 yrs; between years asking it
Shifts with it: 46/108; 43% of analysed shifts

How it is asked

Difficulty, format and skill

Across all 51 questions on Integration by Substitution since 2017.

Difficulty

QUESTIONS

1831

Easy2 · 4%
Medium18 · 35%
Hard31 · 61%

Format

Single correct · 35 Q69%
Numerical value · 16 Q31%

Skill tested

Multi-step · 36 Q71%
Calculation · 14 Q27%
Application · 1 Q2%

Repeats

New idea, or one asked before?

Each column is a year. Blue questions tested an idea no earlier year had asked; navy ones went back to an idea from an earlier year. 2017 is the first year, so everything in it counts as new. Heights are question counts, and Lumi analysed 1 to 24 shifts a year, so compare the split inside a column rather than heights across years.

’17’20’21’22’23’24’25’26

Idea asked for the first time
Idea asked in an earlier year

Coverage of the topic

JEE Main has asked 27 of the 66 ideas in this topic. 34 have never been asked yet.

2734

Asked directly27 · 41%
Touched inside another question5 · 8%
Never asked34 · 52%

Ideas come from Lumi’s JEE syllabus map. A never-asked idea is not a safe skip: across all of JEE Main, 22% of questions since 2024 tested an idea no earlier year had asked.

Drill down

Chapter, idea, question

Every idea JEE Main tested in this topic, with its eight years. Open one to read the questions.

chapterAskedShareYears’17’20’21’22’23’24’25’26

Open50 · 98%98%8/8714571132
27 ideas inside
- OpenRepeated6 · 12%12%3/832
- OpenRepeated5 · 10%10%4/82
- OpenRepeated3 · 6%6%3/8
- OpenRepeated3 · 6%6%3/8
- OpenRepeated3 · 6%6%2/82
- OpenRepeated3 · 6%6%3/8
- Open2 · 4%4%1/82
- OpenRepeated2 · 4%4%2/8
- OpenRepeated2 · 4%4%2/8
- OpenRepeated2 · 4%4%2/8
- OpenRepeated2 · 4%4%2/8
- OpenRepeated2 · 4%4%2/8
- Open1 · 2%2%1/8
- Open1 · 2%2%1/8
- Open1 · 2%2%1/8
- Open1 · 2%2%1/8
- Open1 · 2%2%1/8
- Open1 · 2%2%1/8
- Open1 · 2%2%1/8
- Open1 · 2%2%1/8
- Open1 · 2%2%1/8
- Open1 · 2%2%1/8
- Open1 · 2%2%1/8
- Open1 · 2%2%1/8
- Open1 · 2%2%1/8
- Open1 · 2%2%1/8
- Open1 · 2%2%1/8
Open1 · 2%2%1/8

Squares are the 8 years with analysed shifts, 2017 to 2026. A number is the questions that year when there is more than one; the shade is scaled to one shift, so a year with more analysed shifts (1 to 24 a year) does not look darker. Share is of the level above.

Ideas

Every idea inside this topic

27 different ideas were tested; 12 came more than once.

Composite Function Substitution6×
indefinite integral with radical then evaluate constant
Single correct · 5 shifts · 1 numerical
Fractional Powers LCM Substitution5×
definite integral with fractional powers
Numerical value · 5 shifts · 3 numerical
Exponential Factor Manipulation4×
definite integral with parameter from ap and minimum
Single correct · 4 shifts
Factoring Highest Power for Substitution3×
integrate algebraic radical expression by substitution
Numerical value · 3 shifts · 2 numerical
Numerator-Denominator Derivative Matching3×
integrate rational function of tan then evaluate logs
Single correct · 3 shifts · 1 numerical
Trigonometric Substitution3×
definite integral by trigonometric substitution
Single correct · 3 shifts
Even Power of Secant Substitution2×
definite integral of trigonometric rational expression
Single correct · 2 shifts
Algebraic Manipulation2×
definite integral of rational function of e^x
Numerical value · 2 shifts · 1 numerical
Implicit Derivative Recognition2×
indefinite integral by recognising derivative of xe^x
Single correct · 2 shifts
Twin Trigonometric Integrals Substitution2×
integral with sin x +- cos x substitution
Numerical value · 2 shifts · 1 numerical
Weierstrass Substitution2×
definite integral of rational trig function
Single correct · 2 shifts
Rational Powers Summing to Two Substitution2×
integral of product of linear factor powers
Single correct · 2 shifts
Hyperbolic Substitution1×
integrate powers of sqrt(1+x^2)+x
Numerical value · 1 shift · 1 numerical
Dividing by Cosine Squared Substitution1×
integral by tan substitution then bound on asin x + bcos x
Single correct · 1 shift
Even Secant/Cosecant Power Substitution1×
integral by trigonometric substitution with initial condition
Single correct · 1 shift
Odd Power of Cosine Substitution1×
integrate trig rational function by substitution
Single correct · 1 shift
Odd Power of Cosine Substitution1×
definite integral by substitution of sin x
Numerical value · 1 shift · 1 numerical
Cosine Double Angle Radical Substitution1×
integral of radical rational function with initial value
Single correct · 1 shift
Pure Quadratic Outside and Inside Root Substitution1×
definite integral with nested radical substitution
Numerical value · 1 shift · 1 numerical
General Linear Outside, Quadratic Inside Root Substitution1×
integral of linear over square root of quadratic
Single correct · 1 shift
Logarithmic Derivative Integration Rule1×
integral of ratio with exponential-log terms
Single correct · 1 shift
Negative Even Sum of Trig Powers Substitution1×
integral by tan substitution and coefficient matching
Numerical value · 1 shift · 1 numerical
Quotient Rule Differential Recognition1×
integral recognised as derivative of quotient
Numerical value · 1 shift · 1 numerical
sin(2x) Derivative Substitution1×
indefinite integral with sin2x in root
Single correct · 1 shift
Square Root Derivative Integration Rule1×
evaluate function from derivative via substitution
Numerical value · 1 shift · 1 numerical
Tangent Trigonometric Substitution1×
constants in antiderivative of rational function
Numerical value · 1 shift · 1 numerical
Exact Exponential-Function Derivative Substitution1×
definite integral of exact derivative of product
Single correct · 1 shift

Idea	Times	Shifts	Format	Numerical
Composite Function Substitutionindefinite integral with radical then evaluate constant	6	5	Single correctMulti-step	1
Fractional Powers LCM Substitutiondefinite integral with fractional powers	5	5	Numerical valueMulti-step	3
Exponential Factor Manipulationdefinite integral with parameter from ap and minimum	4	4	Single correctMulti-step	–
Factoring Highest Power for Substitutionintegrate algebraic radical expression by substitution	3	3	Numerical valueMulti-step	2
Numerator-Denominator Derivative Matchingintegrate rational function of tan then evaluate logs	3	3	Single correctMulti-step	1
Trigonometric Substitutiondefinite integral by trigonometric substitution	3	3	Single correctCalculation	–
Even Power of Secant Substitutiondefinite integral of trigonometric rational expression	2	2	Single correctCalculation	–
Algebraic Manipulationdefinite integral of rational function of e^x	2	2	Numerical valueMulti-step	1
Implicit Derivative Recognitionindefinite integral by recognising derivative of xe^x	2	2	Single correctMulti-step	–
Twin Trigonometric Integrals Substitutionintegral with sin x +- cos x substitution	2	2	Numerical valueMulti-step	1
Weierstrass Substitutiondefinite integral of rational trig function	2	2	Single correctCalculation	–
Rational Powers Summing to Two Substitutionintegral of product of linear factor powers	2	2	Single correctCalculation	–
Hyperbolic Substitutionintegrate powers of sqrt(1+x^2)+x	1	1	Numerical valueMulti-step	1
Dividing by Cosine Squared Substitutionintegral by tan substitution then bound on asin x + bcos x	1	1	Single correctMulti-step	–
Even Secant/Cosecant Power Substitutionintegral by trigonometric substitution with initial condition	1	1	Single correctMulti-step	–
Odd Power of Cosine Substitutionintegrate trig rational function by substitution	1	1	Single correctMulti-step	–
Odd Power of Cosine Substitutiondefinite integral by substitution of sin x	1	1	Numerical valueMulti-step	1
Cosine Double Angle Radical Substitutionintegral of radical rational function with initial value	1	1	Single correctMulti-step	–
Pure Quadratic Outside and Inside Root Substitutiondefinite integral with nested radical substitution	1	1	Numerical valueCalculation	1
General Linear Outside, Quadratic Inside Root Substitutionintegral of linear over square root of quadratic	1	1	Single correctCalculation	–
Logarithmic Derivative Integration Ruleintegral of ratio with exponential-log terms	1	1	Single correctMulti-step	–
Negative Even Sum of Trig Powers Substitutionintegral by tan substitution and coefficient matching	1	1	Numerical valueMulti-step	1
Quotient Rule Differential Recognitionintegral recognised as derivative of quotient	1	1	Numerical valueMulti-step	1
sin(2x) Derivative Substitutionindefinite integral with sin2x in root	1	1	Single correctMulti-step	–
Square Root Derivative Integration Ruleevaluate function from derivative via substitution	1	1	Numerical valueCalculation	1
Tangent Trigonometric Substitutionconstants in antiderivative of rational function	1	1	Numerical valueCalculation	1
Exact Exponential-Function Derivative Substitutiondefinite integral of exact derivative of product	1	1	Single correctMulti-step	–

Times and shifts count every analysed shift. Across JEE Main, 78% of questions since 2024 went back to an idea an earlier year had asked; the rest were new. All repeated ideas

Every question

Integration by Substitution questions, year by year

The newest 40 show their opening words as they appeared in the official paper; older ones name the idea they tested. Open one for its options and the official answer.

JEE Main 20262 questions in 2 shifts

5 Apr, Shift 1Q20
The value of the integral $\int_{\pi/6}^{\pi/3}\left(\frac{4 - \operatorname{cosec}^2 x}{\cos^4 x}\right)dx$ is:
Idea: Even Power of Secant Substitution
MediumSingle correct
5 Apr, Shift 2Q18
Let $(2^{1-a}+2^{1+a})$ , $f(a)$ , $(3^a+3^{-a})$ be in A.P. and $\alpha$ be the minimum value of $f(a)$ . Then the value of the integral…
Idea: Exponential Factor Manipulation
HardSingle correct

JEE Main 20253 questions in 3 shifts

3 Apr, Shift 1Q18
Let $f(x) = \int x^3\sqrt{3 - x^2}\,dx$ . If $5f(\sqrt{2}) = -4$ , then $f(1)$ is equal to
Idea: Composite Function Substitution
Asked the year beforeMediumSingle correct
4 Apr, Shift 2Q25
If…
Idea: Hyperbolic Substitution
New ideaHardNumerical value
7 Apr, Shift 2Q24
If…
Idea: Factoring Highest Power for Substitution
HardNumerical value

Questions from the official JEE Main Paper 1 shifts published by NTA (CBSE for 2017). Questions NTA dropped are left out. Topic and idea tags, counts and notes are Lumi’s.

Key findings

Questions a shift, year by year

Difficulty, format and skill

Difficulty

Format

Skill tested

New idea, or one asked before?

Coverage of the topic

Chapter, idea, question

Every idea inside this topic

Integration by Substitution questions, year by year

JEE Main 20262 questions in 2 shifts

JEE Main 20253 questions in 3 shifts

JEE Main 202411 questions in 9 shifts

JEE Main 20237 questions in 7 shifts

JEE Main 20226 questions in 5 shifts

JEE Main 202114 questions in 12 shifts

JEE Main 20207 questions in 7 shifts

JEE Main 20171 question in 1 shift

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