JEE Main · Maths · Integrals

Integration by Parts: JEE Main previous year questions

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

14 questions in the 108 JEE Main shifts Lumi analysed, in 7 of 8 years. Pattern: Asked most years. 3 came in 2024–26. Here is how it is asked and which ideas come back.

Questions: 14; in 13 of 108 shifts
Years asked: 7/8; last asked 2025
Chance in a shift: 19%; calibrated model, how we compute it
Weighted estimate: 0.1; model; history: 0.2 a shift on average, typically 0.1–0.3

Key findings

01
Asked in 7 of 8 years with analysed shifts and in 13 of 108 shifts: 14 questions, #161 of 196 topics. The years that asked it were 1.3 apart on average. Last asked in 2025.
02
62% of its questions since 2020 tested an idea no earlier year had asked; 4 repeated an idea from the year before.
03
The most repeated idea is Algebraic Reduction Formulas, asked 3 times in 3 shifts (2021, 2022, 2023), most often as “relation between parametric integrals via parts”.
04
JEE Main has asked 9 of the 66 ideas in this topic; 49 have never been asked, directly or inside another question.
05
Mostly asked as single correct (79%); 21% need a numerical answer, 0% are easy and 64% hard. It went down from 0.3 to 0.1 questions a shift since 2024.
06
The skill tested most is multi-step (86%), then calculation (14%).

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: 3 in 12 shifts · 2021: 2 in 24 shifts · 2022: 2 in 24 shifts · 2023: 3 in 12 shifts · 2024: 2 in 18 shifts · 2025: 1 in 8 shifts · 2026: 0 in 9 shifts.

Up to 2023: 0.3; a shift, of 25
Since 2024: 0.1; -0.2
Usual gap: 1.3 yrs; between years asking it
Shifts with it: 13/108; 12% of analysed shifts

How it is asked

Difficulty, format and skill

Across all 14 questions on Integration by Parts since 2017.

Difficulty

QUESTIONS

Easy0 · 0%
Medium5 · 36%
Hard9 · 64%

Format

Single correct · 11 Q79%
Numerical value · 3 Q21%

Skill tested

Multi-step · 12 Q86%
Calculation · 2 Q14%

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 9 of the 66 ideas in this topic. 49 have never been asked yet.

9849

Asked directly9 · 14%
Touched inside another question8 · 12%
Never asked49 · 74%

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

Idea, year, question

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

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

OpenRepeated3 · 21%21%3/8
- 11 Apr ’23 S1 · Q26
  For $m, n > 0$ , let $\alpha(m, n) = \int_0^2 t^m(1+3t)^n\,dt$ . If $11\,\alpha(10, 6) + 18\,\alpha(11, 5) = p\,(14)^6$ , then $p$ is equal…
  hardNumerical valueAsked the year before
- 28 Jul ’22 S2 · Q69
  Let $I_n(x)=\int_0^x\frac{1}{(t^2+5)^n}dt,\ n=1,2,3,....$ Then :
  hardSingle correctAsked the year before
- 17 Mar ’21 S2 · Q81
  Let $I_n=\int_1^e x^{19}(\log|x|)^n dx$ , where $n\in\mathbf{N}$ . If $(20)I_{10}=\alpha I_9+\beta I_8$ , for natural numbers $\alpha$ and…
  mediumNumerical valueNew idea
OpenRepeated2 · 14%14%2/8
OpenRepeated2 · 14%14%2/8
OpenRepeated2 · 14%14%2/8
Open1 · 7%7%1/8
Open1 · 7%7%1/8
Open1 · 7%7%1/8
Open1 · 7%7%1/8
Open1 · 7%7%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

9 different ideas were tested; 4 came more than once.

Algebraic Reduction Formulas3×
relation between parametric integrals via parts
Numerical value · 3 shifts · 2 numerical
Logarithm of Quadratic Roots2×
definite integral of log of quadratic root
Single correct · 2 shifts
Trigonometric Reduction Formulas2×
reduction formula for powers of tan
Numerical value · 2 shifts · 1 numerical
Inverse Trig Simplification Before Parts2×
integral of simplified inverse trig function
Single correct · 2 shifts
Roots of Quadratics via Parts1×
integrate radical expression then evaluate logs
Single correct · 1 shift
Generalized Reverse Product Rule1×
integrate expression by recognising derivative
Single correct · 1 shift
Exponential Form with Improper Rational Functions1×
integral of exponential form then higher derivative
Single correct · 1 shift
Exponential Special Form1×
integral of exponential-composite form to find g
Single correct · 1 shift
Integrability over ILATE1×
indefinite integral of trig quotient by parts
Single correct · 1 shift

Idea	Times	Shifts	Format	Numerical
Algebraic Reduction Formulasrelation between parametric integrals via parts	3	3	Numerical valueMulti-step	2
Logarithm of Quadratic Rootsdefinite integral of log of quadratic root	2	2	Single correctCalculation	–
Trigonometric Reduction Formulasreduction formula for powers of tan	2	2	Numerical valueCalculation	1
Inverse Trig Simplification Before Partsintegral of simplified inverse trig function	2	2	Single correctMulti-step	–
Roots of Quadratics via Partsintegrate radical expression then evaluate logs	1	1	Single correctMulti-step	–
Generalized Reverse Product Ruleintegrate expression by recognising derivative	1	1	Single correctMulti-step	–
Exponential Form with Improper Rational Functionsintegral of exponential form then higher derivative	1	1	Single correctMulti-step	–
Exponential Special Formintegral of exponential-composite form to find g	1	1	Single correctMulti-step	–
Integrability over ILATEindefinite integral of trig quotient by parts	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 Parts questions, year by year

The opening words of each question as it appeared in the official paper, with the idea it tested. Open one for its options and the official answer.

JEE Main 20251 question in 1 shift

2 Apr, Shift 2Q18
$4\int_0^1 \left(\frac{1}{\sqrt{3+x^2}+\sqrt{1+x^2}}\right)\mathrm{d}x - 3\log_e(\sqrt{3})$ is equal to :
Idea: Roots of Quadratics via Parts
New ideaHardSingle correct

JEE Main 20242 questions in 2 shifts

4 Apr, Shift 2Q26
If…
Idea: Trigonometric Reduction Formulas
HardNumerical value
9 Apr, Shift 2Q12
The value of the integral $\int_{-1}^{2}\log_e\left(x+\sqrt{x^2+1}\right)dx$ is
Idea: Logarithm of Quadratic Roots
Asked the year beforeMediumSingle correct

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

Idea, year, question

Every idea inside this topic

Integration by Parts questions, year by year

JEE Main 20251 question in 1 shift

JEE Main 20242 questions in 2 shifts

JEE Main 20233 questions in 2 shifts

JEE Main 20222 questions in 2 shifts

JEE Main 20212 questions in 2 shifts

JEE Main 20203 questions in 3 shifts

JEE Main 20171 question in 1 shift

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