JEE Main · Maths · Integrals

Definite Integral Properties: JEE Main previous year questions

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

100 questions in the 108 JEE Main shifts Lumi analysed, in 7 of 8 years. Pattern: Asked most years. 33 came in 2024–26. Here is how it is asked and which ideas come back.

Questions: 100; in 82 of 108 shifts
Years asked: 7/8; last asked 2026
Chance in a shift: 71%; calibrated model, how we compute it
Weighted estimate: 0.8; model; history: 0.7 a shift on average, typically 0.5–0.9

Key findings

01
Asked in 7 of 8 years with analysed shifts and in 82 of 108 shifts: 100 questions, #6 of 196 topics.
02
23% of its questions since 2020 tested an idea no earlier year had asked; 68 repeated an idea from the year before.
03
The most repeated idea is Greatest Integer Function Integrals, asked 29 times in 28 shifts (2020, 2021, 2022, 2023, 2024, 2025, 2026), most often as “definite integral with greatest integer function”.
04
JEE Main has asked 19 of the 56 ideas in this topic; 33 have never been asked, directly or inside another question.
05
Mostly asked as single correct (72%); 28% need a numerical answer, 7% are easy and 42% hard. It went up from 0.6 to 0.8 questions a shift since 2024.
06
The skill tested most is multi-step (57%), then calculation (32%).

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: 0 in 1 shift · 2020: 8 in 12 shifts · 2021: 31 in 24 shifts · 2022: 21 in 24 shifts · 2023: 7 in 12 shifts · 2024: 19 in 18 shifts · 2025: 7 in 8 shifts · 2026: 7 in 9 shifts.

Up to 2023: 0.6; a shift, of 25
Since 2024: 0.8; 0.3
Usual gap: 1.0 yrs; between years asking it
Shifts with it: 82/108; 76% of analysed shifts

Recent framings

definite integral with exponential denominator trick
integrate rational function then evaluate logs
numerical: improper definite integral value

Revise first

Greatest Integer Function Integrals · asked 2020, 2021, 2022, 2023, 2024, 2025, 2026
King's Property · asked 2020, 2021, 2022, 2023, 2024, 2025
Symmetric Limits with Exponential Denominator Trick · asked 2021, 2022, 2024, 2025, 2026

How it is asked

Difficulty, format and skill

Across all 100 questions on Definite Integral Properties since 2017.

Difficulty

QUESTIONS

5142

Easy7 · 7%
Medium51 · 51%
Hard42 · 42%

Format

Single correct · 72 Q72%
Numerical value · 28 Q28%

Skill tested

Multi-step · 57 Q57%
Calculation · 32 Q32%
Application · 11 Q11%

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 19 of the 56 ideas in this topic. 33 have never been asked yet.

1933

Asked directly19 · 34%
Touched inside another question4 · 7%
Never asked33 · 59%

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

OpenRepeated29 · 29%29%7/81183222
- 2 Apr ’26 S1 · Q19
  Let $[\cdot]$ denote the greatest integer function. Then the value of $\int_0^3\left(\frac{e^x+e^{-x}}{[x]!}\right)dx$ is :
  mediumSingle correctAsked the year before
- 4 Apr ’26 S1 · Q19
  Let $\int_{-2}^{2}\left(|\sin x|+[x\sin x]\right)dx=2(3-\cos 2)+\beta$ , where $[\cdot]$ is the greatest integer function. Then…
  hardSingle correctAsked the year before
- 2 Apr ’25 S1 · Q23
  Let $[\cdot]$ 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 valueAsked the year before
- 3 Apr ’25 S1 · Q19
  Let the domain of the function $f(x) = \log_2 \log_4 \log_6 (3 + 4x - x^2)$ be $(a, b)$ . If…
  hardSingle correctAsked the year before
- 6 Apr ’24 S2 · Q25
  Let $[t]$ denote the largest integer less than or equal to $t$ . If…
  hardNumerical valueAsked the year before
- 30 Jan ’24 S1 · Q25
  The value of $9\int_0^9\left[\sqrt{\frac{10x}{x+1}}\right]dx$ , where $[t]$ denotes the greatest integer less than or equal to $t$ , is…
  hardNumerical valueAsked the year before
- 8 Apr ’23 S1 · Q24
  Read the question
  hardNumerical valueAsked the year before
- 8 Apr ’23 S2 · Q26
  Read the question
  mediumNumerical valueAsked the year before
- 11 Apr ’23 S2 · Q11
  Read the question
  hardSingle correctAsked the year before
- 25 Jul ’22 S2 · Q9
  Read the question
  mediumSingle correctAsked the year before
- 27 Jul ’22 S1 · Q7
  Read the question
  mediumSingle correctAsked the year before
- 27 Jul ’22 S2 · Q9
  Read the question
  mediumSingle correctAsked the year before
- 27 Jul ’22 S2 · Q27
  Read the question
  hardNumerical valueAsked the year before
- 29 Jul ’22 S2 · Q5
  Read the question
  hardSingle correctAsked the year before
- 27 Jun ’22 S2 · Q9
  Read the question
  hardSingle correctAsked the year before
- 28 Jun ’22 S1 · Q7
  Read the question
  hardSingle correctAsked the year before
- 29 Jun ’22 S1 · Q17
  Read the question
  mediumSingle correctAsked the year before
- 3 Aug ’21 S1 · Q70
  Read the question
  mediumSingle correctAsked the year before
- 3 Aug ’21 S2 · Q84
  Read the question
  mediumNumerical valueAsked the year before
- 26 Aug ’21 S2 · Q74
  Read the question
  mediumSingle correctAsked the year before
- 31 Aug ’21 S1 · Q84
  Read the question
  mediumNumerical valueAsked the year before
- 31 Aug ’21 S2 · Q69
  Read the question
  hardSingle correctAsked the year before
- 24 Feb ’21 S1 · Q87
  Read the question
  mediumNumerical valueAsked the year before
- 24 Feb ’21 S2 · Q65
  Read the question
  mediumSingle correctAsked the year before
- 25 Feb ’21 S1 · Q68
  Read the question
  mediumSingle correctAsked the year before
- 16 Mar ’21 S2 · Q74
  Read the question
  mediumSingle correctAsked the year before
- 17 Mar ’21 S1 · Q85
  Read the question
  hardNumerical valueAsked the year before
- 17 Mar ’21 S2 · Q66
  Read the question
  hardSingle correctAsked the year before
- 2 Sep ’20 S2 · Q74
  Read the question
  mediumNumerical valueNew idea
OpenRepeated18 · 18%18%6/824272
OpenRepeated10 · 10%10%5/8233
OpenRepeated6 · 6%6%3/823
OpenRepeated6 · 6%6%3/84
OpenRepeated6 · 6%6%4/83
OpenRepeated4 · 4%4%3/82
OpenRepeated4 · 4%4%3/82
OpenRepeated3 · 3%3%2/82
OpenRepeated2 · 2%2%2/8
OpenRepeated2 · 2%2%2/8
OpenRepeated2 · 2%2%2/8
OpenRepeated2 · 2%2%2/8
Open1 · 1%1%1/8
Open1 · 1%1%1/8
Open1 · 1%1%1/8
Open1 · 1%1%1/8
Open1 · 1%1%1/8
Open1 · 1%1%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

19 different ideas were tested; 13 came more than once.

Greatest Integer Function Integrals29×
definite integral with greatest integer function
Single correct · 28 shifts · 11 numerical
King's Property18×
definite integral using king's property
Single correct · 18 shifts · 4 numerical
Symmetric Limits with Exponential Denominator Trick10×
definite integral using king's property
Single correct · 10 shifts · 1 numerical
Periodic Function Property6×
definite integral over multiple periods
Single correct · 6 shifts
Modulus Function Interval Splitting6×
definite integral with modulus splitting
Numerical value · 6 shifts · 3 numerical
Reduction Formulae in Definite Integrals6×
definite integral with reduction relation
Single correct · 6 shifts · 1 numerical
Even and Odd Functions4×
integrate rational function then evaluate logs
Single correct · 4 shifts · 1 numerical
Bounding and Estimation Property4×
bound a definite integral
Single correct · 4 shifts
Fractional Part Function Integrals3×
definite integral with fractional part function
Single correct · 3 shifts · 1 numerical
Reciprocal Substitution (x = 1/t)2×
improper definite integral value
Single correct · 2 shifts
Removing 'x' using King's Property2×
definite integral using king's property
Numerical value · 2 shifts · 1 numerical
Beta Function Integrals2×
express definite integral as beta function
Numerical value · 2 shifts · 1 numerical
Interval Splitting2×
definite integral of piecewise function with modulus
Numerical value · 2 shifts · 1 numerical
Integral of Inverse Functions1×
definite integral sum of inverse functions
Numerical value · 1 shift · 1 numerical
Integrals of Min and Max Functions1×
definite integral of max of modulus functions
Numerical value · 1 shift · 1 numerical
Max/Min Function Integrals1×
definite integral with max of two functions
Numerical value · 1 shift · 1 numerical
Reciprocal Limits Substitution Trick1×
sum of integral function at reciprocal arguments
Single correct · 1 shift
King's Property Rational Shortcut1×
definite integral by king property
Single correct · 1 shift
Geometric Area for Piecewise Linear Integrals1×
definite integral of composed modulus function
Single correct · 1 shift

Idea	Times	Shifts	Format	Numerical
Greatest Integer Function Integralsdefinite integral with greatest integer function	29	28	Single correctCalculation	11
King's Propertydefinite integral using king's property	18	18	Single correctMulti-step	4
Symmetric Limits with Exponential Denominator Trickdefinite integral using king's property	10	10	Single correctMulti-step	1
Periodic Function Propertydefinite integral over multiple periods	6	6	Single correctMulti-step	–
Modulus Function Interval Splittingdefinite integral with modulus splitting	6	6	Numerical valueCalculation	3
Reduction Formulae in Definite Integralsdefinite integral with reduction relation	6	6	Single correctMulti-step	1
Even and Odd Functionsintegrate rational function then evaluate logs	4	4	Single correctMulti-step	1
Bounding and Estimation Propertybound a definite integral	4	4	Single correctApplication	–
Fractional Part Function Integralsdefinite integral with fractional part function	3	3	Single correctMulti-step	1
Reciprocal Substitution (x = 1/t)improper definite integral value	2	2	Single correctCalculation	–
Removing 'x' using King's Propertydefinite integral using king's property	2	2	Numerical valueMulti-step	1
Beta Function Integralsexpress definite integral as beta function	2	2	Numerical valueMulti-step	1
Interval Splittingdefinite integral of piecewise function with modulus	2	2	Numerical valueCalculation	1
Integral of Inverse Functionsdefinite integral sum of inverse functions	1	1	Numerical valueMulti-step	1
Integrals of Min and Max Functionsdefinite integral of max of modulus functions	1	1	Numerical valueCalculation	1
Max/Min Function Integralsdefinite integral with max of two functions	1	1	Numerical valueMulti-step	1
Reciprocal Limits Substitution Tricksum of integral function at reciprocal arguments	1	1	Single correctMulti-step	–
King's Property Rational Shortcutdefinite integral by king property	1	1	Single correctApplication	–
Geometric Area for Piecewise Linear Integralsdefinite integral of composed modulus function	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

Definite Integral Properties 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.

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

Definite Integral Properties questions, year by year

JEE Main 20267 questions in 6 shifts

JEE Main 20257 questions in 6 shifts

JEE Main 202419 questions in 14 shifts

JEE Main 20237 questions in 7 shifts

JEE Main 202221 questions in 18 shifts

JEE Main 202131 questions in 23 shifts

JEE Main 20208 questions in 8 shifts

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