JEE (Main) · Mathematics, Class 12
Integral Calculus: JEE Main PYQ, 2020–2025
Integral Calculus (Mathematics, Class 12) has been asked 312 times in the 111 JEE (Main) shift papers from 2020–2025: about 2.8 questions a paper, 10% of all Mathematics questions. It is the most-asked of 19 Mathematics chapters. The 2025 shifts had 42 from it in all. Last checked 28 October 2026.
Questions per year, all shifts
Year by year.
| Year | Questions |
|---|---|
| 2020 | 31in 14 papers |
| 2021 | 75in 26 papers |
| 2022 | 68in 24 papers |
| 2023 | 36in 12 papers |
| 2024 | 60in 18 papers |
| 2025 | 42in 17 papers |
JEE (Main) is sat in many shifts a year, each with its own paper; the counts add up every shift. 2021–2024 papers had 30 questions a subject, the others 25.
Inside the chapter
Topics asked most.
- Area under simple curves96
- Properties of definite integrals95
- Integration by substitution45
- Fundamental theorem of calculus40
- Integration by parts21
- Integral as an antiderivative9
- Partial fractions6
Try it
10 Integral Calculus questions to practise.
Our own questions, written in the JEE Main pattern and checked, each with an explanation. Pick an option to see the answer.
Q1 · Mathematics · Integral Calculus
What is the value of the integral of x/√(x² + 9) with respect to x from x = 0 to x = 4?
Answer: (B)
With u = x² + 9, du = 2x dx, and u runs from 9 to 25. The integral is (1/2)∫u^(−1/2) du = [√u] from 9 to 25 = 5 − 3 = 2.
Q2 · Mathematics · Integral Calculus
Find the area of the region bounded by the curve y = eˣ, the line y = e² and the y-axis.
Answer: (C)
The curve meets y = e² at x = 2, and at x = 0 it meets the y-axis at y = 1. For 0 ≤ x ≤ 2 the line y = e² is above the curve, so Area = ∫₀²(e² − eˣ)dx = [e²x − eˣ]₀² = (2e² − e²) − (0 − 1) = e² + 1. Check by horizontal strips: ∫₁^{e²} ln y dy = [y ln y − y]₁^{e²} = e² + 1.
Q3 · Mathematics · Integral Calculus
The value of ∫ from x = e^e to x = e^(e²) of dx / [x · ln x · ln(ln x)] is
Answer: (A)
The factors x, ln x and ln(ln x) in the denominator suggest a nested logarithmic substitution. Put w = ln(ln x), so dw = dx/(x ln x). The integral becomes ∫dw/w. At x = e^e, ln x = e, so w = ln e = 1. At x = e^(e²), ln x = e², so w = ln(e²) = 2. Hence the value is [ln w]₁² = ln 2 − ln 1 = ln 2.
Q4 · Mathematics · Integral Calculus
If (5x + 7)/((x + 1)(x + 3)) = A/(x + 1) + B/(x + 3) for all x other than −1 and −3, find the value of B.
Answer: (A)
Multiply through by (x+1)(x+3) to get 5x + 7 = A(x+3) + B(x+1). Put x = −3: −15 + 7 = −8 = B(−2), so B = 4. Put x = −1: 2 = 2A, so A = 1. Check: (x+3) + 4(x+1) = 5x + 7.
Q5 · Mathematics · Integral Calculus
Evaluate ∫ x sec²x dx.
Answer: (A)
Choose u = x (algebraic) and dv = sec²x dx, so v = tan x. Then ∫x sec²x dx = x tan x − ∫tan x dx. Now ∫tan x dx = −ln|cos x| + C, so the result is x tan x + ln|cos x| + C. Differentiating gives tan x + x sec²x − tan x = x sec²x, which confirms it.
Q6 · Mathematics · Integral Calculus
Evaluate ∫ (x² + 1)/(x⁴ + 1) dx for x > 0, where C is the constant of integration.
Answer: (A)
Divide numerator and denominator by x²: the integrand becomes (1 + 1/x²)/(x² + 1/x²). Put t = x − 1/x, so dt = (1 + 1/x²)dx. Also x² + 1/x² = t² + 2. The integral becomes ∫dt/(t² + 2) = (1/√2) tan⁻¹(t/√2) + C. Substituting back, t/√2 = (x² − 1)/(√2 x).
Q7 · Mathematics · Integral Calculus
What is the area of the region in the first quadrant bounded by the circle x² + y² = 4 and the coordinate axes?
Answer: (C)
The circle x² + y² = 4 has radius 2. The first-quadrant region is a quarter circle. Using integration, ∫₀² √(4 − x²) dx = [(x/2)√(4 − x²) + 2 sin⁻¹(x/2)]₀² = 2·(π/2) = π square units.
Q8 · Mathematics · Integral Calculus
Evaluate the indefinite integral of e^(√x)/√x with respect to x, for x > 0. (C is the constant of integration.)
Answer: (A)
Let u = √x. Then du = dx/(2√x), so dx/√x = 2 du. The integral becomes 2∫e^u du = 2e^u + C = 2e^(√x) + C. Differentiating back gives 2·e^(√x)·1/(2√x) = e^(√x)/√x, which confirms the result.
Q9 · Mathematics · Integral Calculus
Let f(x) = x³ + x for real x, and let g be the inverse function of f. Find the area of the region bounded by the curve y = g(x), the x-axis and the line x = 2, in square units.
Answer: (A)
f is strictly increasing with f(0) = 0 and f(1) = 2. So g(0) = 0 and g(2) = 1, and the region is the area under y = g(x) for 0 ≤ x ≤ 2. Reflect it in y = x. The region under the graph of g, read as x as a function of y, lies between the curve y = f(x) and the y-axis. Equivalently, the area under g on [0, 2] plus the area under f on [0, 1] equals the rectangle 1 × 2 = 2. So ∫₀² g(x) dx = 2 − ∫₀¹ (x³ + x) dx = 2 − (1/4 + 1/2) = 5/4. No formula for g is needed.
Q10 · Mathematics · Integral Calculus
If ∫ f(x) dx = x³ + sin x + C, what is the value of f(0)?
Answer: (A)
If F(x) = ∫f(x) dx, then F′(x) = f(x). So f(x) = d/dx(x³ + sin x + C) = 3x² + cos x. Then f(0) = 3(0) + cos 0 = 1.
Free with an account
All 312 real JEE Main questions from Integral Calculus.
The questions from the actual papers, with the official answer and an explanation for each — chapter by chapter, or as full timed papers.
Solve them freeQuestions
About Integral Calculus in JEE Main.
How many questions come from Integral Calculus in JEE Main?
Integral Calculus (Mathematics, Class 12) has been asked 312 times in the 111 JEE (Main) shift papers from 2020–2025: about 2.8 questions a paper, 10% of all Mathematics questions. It is the most-asked of 19 Mathematics chapters. The 2025 shifts had 42 from it in all.
Which topics of Integral Calculus are asked most in JEE Main?
Area under simple curves (96), Properties of definite integrals (95), Integration by substitution (45), Fundamental theorem of calculus (40), counted over the papers from 2020–2025.
Is Integral Calculus important for JEE Main 2027?
It is in the syllabus in force, so it can be asked. By past papers it ranks 1 of 19 in Mathematics; use that to decide how much time it gets, not whether to study it.
Where can I solve Integral Calculus JEE Main previous year questions?
On Nikaalle, free: sign up and practise every past JEE Main question from Integral Calculus with an explanation for each, or sit the full papers timed. The ten questions on this page are our own, written in the same pattern.
Your JEE Main 2027 plan, chapter by chapter.
A week-wise plan that gives the heavy chapters their time. Free, from a non-profit trust.