How much of JEE Main comes from Integral Calculus?

JEEGenie estimates about 8% of JEE Main marks come from Integral Calculus, based on recent papers. The share varies from year to year.

Which formulas matter most in Integral Calculus?

The Integral Calculus sheet on this page has 23 formula entries, starting with Indefinite Integral, Definite Integral, Power Rule for Integration, Integration of Trigonometric Functions, Integration of Exponential and Logarithmic Functions. Each one comes with when to use it and the trap to avoid.

What topics does Integral Calculus cover for JEE?

Integral Calculus covers Indefinite Integration, Methods of Integration, Definite Integrals, Properties of Definite Integrals, Area Under Curves, Reduction Formulae.

How should I revise Integral Calculus before the exam?

Learn each formula together with its condition of use, read the trap listed beside it, then practise questions and review every mistake. The practice link on this page opens Integral Calculus questions on JEEGenie.

How many previous-year questions are there on Integral Calculus?

JEEGenie has 1010 previous-year questions on Integral Calculus, and this page shows a few with worked explanations.

Is practice on Integral Calculus free on JEEGenie?

Sign-up is free and includes a 7-day Pro trial. After the trial, free accounts keep a limited daily quota.

Integral Calculus for JEE Main & Advanced — formulas, what is asked, and traps

JEEGenie estimates about 8% of JEE Main marks come from Integral Calculus. This page covers 23 formula entries, each with when to use it, what JEE asks and the traps. A common slip: Forgetting the constant of integration C in indefinite integrals.

Key points to know

Indefinite Integral. The indefinite integral of a function f(x) is the family of all antiderivatives, denoted ∫f(x)dx = F(x) + C, where F'(x) = f(x) and C is the constant of integration.
Definite Integral. The definite integral ∫_a^b f(x)dx represents the signed area under the curve y=f(x) from x=a to x=b, and is a number (no constant of integration).

Formula sheet

Name	Formula	When to use
Indefinite Integral	$\int f (x) d x = F (x) + C$	When finding the general antiderivative of a function without specified limits.
Definite Integral	$\int_{a}^{b} f (x) d x = F (b) - F (a)$	When computing area, net change, or evaluating integrals with limits.
Power Rule for Integration	$\int x^{n} d x = \frac{x^{n + 1}}{n + 1} + C (n \neq - 1)$ $\int \frac{1}{x} d x = \ln ∣ x ∣ + C$	When integrating polynomial terms or any x^n form.
Integration of Trigonometric Functions	$\int \sin x d x = - \cos x + C$ $\int \cos x d x = \sin x + C$ $\int \sec^{2} x d x = \tan x + C$ $\int \csc^{2} x d x = - \cot x + C$ $\int \sec x \tan x d x = \sec x + C$ $\int \csc x \cot x d x = - \csc x + C$	When integrating trigonometric functions directly or after substitution.
Integration of Exponential and Logarithmic Functions	$\int e^{x} d x = e^{x} + C$ $\int a^{x} d x = \frac{a^{x}}{\ln a} + C$ $\int \ln x d x = x \ln x - x + C$	When integrating exponential or logarithmic functions directly.
Integration by Substitution	$\int f (g (x)) g^{'} (x) d x = \int f (u) d u$ where $u = g (x)$	When the integrand contains a function and its derivative (or a constant multiple).
Integration by Parts	$\int u d v = u v - \int v d u$	When the integrand is a product of two different types of functions (e.g., x e^x, x sin x, ln x).
Integration by Partial Fractions	For distinct linear factors: $\frac{A}{x - a} + \frac{B}{x - b}$ For repeated linear: $\frac{A}{x - a} + \frac{B}{(x - a)^{2}}$ For quadratic: $\frac{A x + B}{x^{2} + p x + q}$	When integrating rational functions where denominator can be factored.
Standard Integrals Involving Quadratic Expressions	$\int \frac{d x}{x^{2} + a^{2}} = \frac{1}{a} \arctan \frac{x}{a} + C$ $\int \frac{d x}{\sqrt{a^{2} - x^{2}}} = \arcsin \frac{x}{a} + C$ $\int \frac{d x}{\sqrt{x^{2} \pm a^{2}}} = \ln ∣ x + \sqrt{x^{2} \pm a^{2}} ∣ + C$	When the integrand involves $\sqrt{a^{2} - x^{2}}$ , $\sqrt{x^{2} \pm a^{2}}$ , or $\frac{1}{x^{2} + a^{2}}$ .
Integration Using Trigonometric Identities	$\sin^{2} x = \frac{1 - \cos 2 x}{2}$ $\cos^{2} x = \frac{1 + \cos 2 x}{2}$ $\sin x \cos x = \frac{1}{2} \sin 2 x$	When integrating powers of sin and cos, or products like sin mx cos nx.
Integration of Irrational Functions	For $\sqrt{a^{2} - x^{2}}$ : $x = a \sin θ$ For $\sqrt{x^{2} + a^{2}}$ : $x = a \tan θ$ For $\sqrt{x^{2} - a^{2}}$ : $x = a \sec θ$	When the integrand contains $\sqrt{a^{2} - x^{2}}$ , $\sqrt{x^{2} \pm a^{2}}$ , or $\sqrt{a x + b}$ .
Definite Integral Properties	$\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x$ $\int_{0}^{a} f (x) d x = \int_{0}^{a} f (a - x) d x$ Even: $\int_{- a}^{a} f (x) d x = 2 \int_{0}^{a} f (x) d x$ Odd: $\int_{- a}^{a} f (x) d x = 0$	When evaluating definite integrals, especially with symmetric limits or periodic functions.
Leibniz Rule for Differentiation Under Integral Sign	$\frac{d}{d t} \int_{a (t)}^{b (t)} f (x, t) d x = f (b (t), t) b^{'} (t) - f (a (t), t) a^{'} (t) + \int_{a (t)}^{b (t)} \frac{\partial f}{\partial t} d x$	When evaluating integrals that depend on a parameter, especially in JEE advanced problems.
Area Under Curves	Area = $\int_{a}^{b} ∣ f (x) - g (x) ∣ d x$	When computing area between curves or under a curve.
Integration as Limit of a Sum	$\int_{a}^{b} f (x) d x = \lim_{n \to \infty} \frac{b - a}{n} \sum_{r = 1}^{n} f (a + r \frac{b - a}{n})$	When evaluating limits of sums that resemble Riemann sums.
Gamma Function and Beta Function	$Γ (n) = \int_{0}^{\infty} x^{n - 1} e^{- x} d x$ $B (m, n) = \int_{0}^{1} x^{m - 1} (1 - x)^{n - 1} d x = \frac{Γ (m) Γ (n)}{Γ (m + n)}$	When evaluating integrals involving powers and exponentials, or products of powers.
Reduction Formulas	$\int \sin^{n} x d x = - \frac{1}{n} \sin^{n - 1} x \cos x + \frac{n - 1}{n} \int \sin^{n - 2} x d x$ $\int \cos^{n} x d x = \frac{1}{n} \cos^{n - 1} x \sin x + \frac{n - 1}{n} \int \cos^{n - 2} x d x$	When integrating high powers of trigonometric functions.
Integration of Piecewise Functions	$\int_{a}^{b} f (x) d x = \int_{a}^{c} f_{1} (x) d x + \int_{c}^{b} f_{2} (x) d x$ where $f (x) = {\begin{cases} f_{1} (x) & x < c \\ f_{2} (x) & x \geq c \end{cases}$	When the integrand is defined piecewise, e.g., \|x\|, floor functions, or mod functions.
Integration Using Euler's Substitution	If $a > 0$ : $\sqrt{a x^{2} + b x + c} = t - \sqrt{a} x$ If $c > 0$ : $\sqrt{a x^{2} + b x + c} = x t + \sqrt{c}$	When integrating rational functions with √(ax^2+bx+c) that cannot be handled by standard trigonometric substitutions.
Integration of Rational Functions of sin x and cos x	$t = \tan \frac{x}{2}$ $\sin x = \frac{2 t}{1 + t^{2}}$ $\cos x = \frac{1 - t^{2}}{1 + t^{2}}$ $d x = \frac{2}{1 + t^{2}} d t$	When integrating rational functions of sin x and cos x, e.g., ∫ dx/(a sin x + b cos x + c).
Integration of Irrational Functions of the Form ∫dx/√(ax^2+bx+c)	$\int \frac{d x}{\sqrt{a x^{2} + b x + c}} = \frac{1}{\sqrt{a}} \ln ∣ 2 a x + b + 2 \sqrt{a (a x^{2} + b x + c)} ∣ + C$ (for a>0)	When integrand is 1/√(quadratic) or similar.
Integration of √(ax^2+bx+c)	$\int \sqrt{a^{2} - x^{2}} d x = \frac{x}{2} \sqrt{a^{2} - x^{2}} + \frac{a^{2}}{2} \arcsin \frac{x}{a} + C$ $\int \sqrt{x^{2} \pm a^{2}} d x = \frac{x}{2} \sqrt{x^{2} \pm a^{2}} \pm \frac{a^{2}}{2} \ln ∣ x + \sqrt{x^{2} \pm a^{2}} ∣ + C$	When integrating √(quadratic) expressions.
Integration Using Special Substitutions (e.g., x = a cos^2 θ + b sin^2 θ)	$x = a \cos^{2} θ + b \sin^{2} θ$ Then $d x = 2 (b - a) \sin θ \cos θ d θ$	When integrand contains √((x-a)(b-x)) or similar forms.

Open the Integral Calculus formula page

What JEE asks

Indefinite Integral. JEE often tests basic antiderivatives and the need to include C in indefinite integrals.
Definite Integral. JEE uses definite integrals for area problems and evaluating limits of sums.
Power Rule for Integration. JEE tests this as a basic building block in almost every integral problem.
Integration of Trigonometric Functions. JEE often uses these in combination with substitution or integration by parts.
Integration of Exponential and Logarithmic Functions. JEE tests these in definite integrals and area problems.
Integration by Substitution. JEE often uses substitution to reduce integrals to standard forms, especially in trigonometric and exponential functions.
Integration by Parts. JEE tests this with products like x^n e^x, x^n sin x, or inverse trigonometric functions.
Integration by Partial Fractions. JEE often uses partial fractions with linear and quadratic factors, sometimes with substitution.
Standard Integrals Involving Quadratic Expressions. JEE tests these after completing the square or substitution.
Integration Using Trigonometric Identities. JEE uses these to simplify integrals of trigonometric powers.
Integration of Irrational Functions. JEE tests these substitutions in definite integrals and area problems.
Definite Integral Properties. JEE often uses these properties to simplify integrals, especially the f(a-x) property.
Leibniz Rule for Differentiation Under Integral Sign. JEE uses this to evaluate tricky definite integrals by introducing a parameter.
Area Under Curves. JEE often asks area bounded by curves, requiring careful identification of limits and absolute values.
Integration as Limit of a Sum. JEE tests this in limit problems where sum is converted to integral.
Gamma Function and Beta Function. JEE advanced may use these in definite integrals with limits 0 to ∞ or 0 to 1.
Reduction Formulas. JEE tests reduction formulas for sin^n x, cos^n x, and sometimes for sec^n x.
Integration of Piecewise Functions. JEE often tests integrals of |x-a| or |sin x| over intervals.
Integration Using Euler's Substitution. JEE advanced may use Euler substitutions for complex integrals.
Integration of Rational Functions of sin x and cos x. JEE tests this substitution for integrals that are otherwise hard to solve.
Integration of Irrational Functions of the Form ∫dx/√(ax^2+bx+c). JEE often uses this after completing the square.
Integration of √(ax^2+bx+c). JEE tests these standard forms directly or after substitution.
Integration Using Special Substitutions (e.g., x = a cos^2 θ + b sin^2 θ). JEE advanced may use this for tricky definite integrals.

Common traps

Forgetting the constant of integration C in indefinite integrals. Indefinite Integral
Forgetting to subtract F(a) from F(b) (order matters). Definite Integral
Applying power rule for n = -1 (gives ln|x|, not x^0/0). Power Rule for Integration
Forgetting the negative sign in ∫sin x dx and ∫csc^2 x dx. Integration of Trigonometric Functions
Forgetting the division by ln a for a^x. Integration of Exponential and Logarithmic Functions
Forgetting to change dx to du correctly. Integration by Substitution
Choosing u and dv incorrectly (e.g., picking dv that is hard to integrate). Integration by Parts
Not checking if degree of numerator ≥ denominator (perform polynomial division first). Integration by Partial Fractions
Forgetting the factor 1/a in arctan form. Standard Integrals Involving Quadratic Expressions
Not applying identities to reduce powers. Integration Using Trigonometric Identities
Forgetting to convert dx in terms of dθ. Integration of Irrational Functions
Forgetting the sign change when swapping limits. Definite Integral Properties
Forgetting to differentiate the limits. Leibniz Rule for Differentiation Under Integral Sign
Not taking absolute value when curves cross. Area Under Curves

Topics in Integral Calculus

Indefinite Integration
Methods of Integration
Definite Integrals
Properties of Definite Integrals
Area Under Curves
Reduction Formulae

Previous-year questions on Integral Calculus

JEEGenie has 1010 previous-year questions on Integral Calculus. A few worked examples follow.

Year	Questions
2026	57
2025	52
2024	79
2023	78
2022	73
2021	89
2020	46
2019	106
2018	32
2017	21
2016	21
2015	28
2014	30
2013	18
2012	19
2011	9
2010	6
2009	7
2008	8
2007	8
2006	14
2005	16
2004	18
2003	11
2002	17
2001	5
2000	7
1999	9
1998	5
1997	15
1996	8
1995	12
1994	4
1993	6
1992	4
1991	5
1990	7
1989	7
1988	8
1987	5
1986	4
1985	5
1984	5
1983	6
1982	4
1981	6
1980	2
1979	4
1978	4

Question 1 medium

Let [.] denote the greatest integer function. Then the value of

\int_{0}^{3} (\frac{e^{x} + e^{- x}}{[x]!}) d x

is :

$e^{2} + e^{3} - \frac{1}{e^{2}} - \frac{1}{e^{3}}$
$\frac{1}{2} (e^{2} + e^{3} - \frac{1}{e^{2}} - \frac{1}{e^{3}})$
$e^{2} + e^{3} - \frac{1}{2 e^{2}} - \frac{1}{2 e^{3}}$
$\frac{1}{2} (e^{2} + e^{3}) - \frac{1}{e^{2}} - \frac{1}{e^{3}}$

Answer: B

Explanation. Since

[x]

is constant on intervals

[0, 1)

[1, 2)

[2, 3)

, we split the integral:

I = \int_{0}^{1} \frac{e^{x} + e^{- x}}{0!} d x + \int_{1}^{2} \frac{e^{x} + e^{- x}}{1!} d x + \int_{2}^{3} \frac{e^{x} + e^{- x}}{2!} d x

Since

0! = 1

1! = 1

2! = 2

I = \int_{0}^{1} (e^{x} + e^{- x}) d x + \int_{1}^{2} (e^{x} + e^{- x}) d x + \frac{1}{2} \int_{2}^{3} (e^{x} + e^{- x}) d x

Combine the first two integrals:

\int_{0}^{2} (e^{x} + e^{- x}) d x = {[e^{x} - e^{- x}]}_{0}^{2} = (e^{2} - e^{- 2}) - (1 - 1) = e^{2} - e^{- 2}

Third integral:

\frac{1}{2} \int_{2}^{3} (e^{x} + e^{- x}) d x = \frac{1}{2} {[e^{x} - e^{- x}]}_{2}^{3} = \frac{1}{2} ((e^{3} - e^{- 3}) - (e^{2} - e^{- 2}))

Thus:

I = (e^{2} - e^{- 2}) + \frac{1}{2} (e^{3} - e^{- 3} - e^{2} + e^{- 2})

Simplify:

I = e^{2} - e^{- 2} + \frac{1}{2} e^{3} - \frac{1}{2} e^{- 3} - \frac{1}{2} e^{2} + \frac{1}{2} e^{- 2} = \frac{1}{2} e^{2} + \frac{1}{2} e^{3} - \frac{1}{2} e^{- 2} - \frac{1}{2} e^{- 3}

I = \frac{1}{2} (e^{2} + e^{3} - \frac{1}{e^{2}} - \frac{1}{e^{3}})

B

From NTA/IIT papers: JEE Main 2026 · 2 Apr Shift 1

Question 2 easy

The area of the region enclosed by the parabolas

y = x^{2} - 5 x

and

y = 7 x - x^{2}

is ________.

Answer: 72

Explanation. Two parabolas bound a finite region between their points of intersection, so first find where they meet:

x^{2} - 5 x = 7 x - x^{2} ⟹ 2 x^{2} - 12 x = 0 ⟹ x = 0 or x = 6.

Between these limits decide which curve is on top. At

x = 3

y = 7 (3) - 9 = 12

for the second parabola and

y = 9 - 15 = - 6

for the first, so

y = 7 x - x^{2}

is the upper curve on

(0, 6)

. The area is the integral of (upper

-

lower):

A = \int_{0}^{6} [(7 x - x^{2}) - (x^{2} - 5 x)] d x = \int_{0}^{6} (12 x - 2 x^{2}) d x .

Integrating,

A = [6 x^{2} - \frac{2 x^{3}}{3}]_{0}^{6} = 6 (36) - \frac{2 (216)}{3} = 216 - 144 = 72.

The enclosed area is

72

square units.

From NTA/IIT papers: JEE Main 2024 · 5 Apr Shift 1

Question 3 medium

Let

5 f (x) + 4 f (\frac{1}{x}) = \frac{1}{x} + 3, x > 0

. Then

18 \int_{1}^{2} f (x) d x

is equal to :

$10 \log_{e} 2 + 6$
$5 \log_{e} 2 - 3$
$10 \log_{e} 2 - 6$
$5 \log_{e} 2 + 3$

Answer: C

Explanation. Replace

x

\frac{1}{x}

5 f (\frac{1}{x}) + 4 f (x) = x + 3.

Multiply original equation by 5 and this new equation by 4:

25 f (x) + 20 f (\frac{1}{x}) = \frac{5}{x} + 15,

20 f (\frac{1}{x}) + 16 f (x) = 4 x + 12.

Subtract:

9 f (x) = \frac{5}{x} + 15 - 4 x - 12 = \frac{5}{x} - 4 x + 3.

Thus

f (x) = \frac{1}{9} (\frac{5}{x} - 4 x + 3) .

Now integrate:

\int_{1}^{2} f (x) d x = \frac{1}{9} {[5 \ln x - 2 x^{2} + 3 x]}_{1}^{2} .

Evaluate:

= \frac{1}{9} [(5 \ln 2 - 8 + 6) - (0 - 2 + 3)] = \frac{1}{9} (5 \ln 2 - 2 - 1) = \frac{1}{9} (5 \ln 2 - 3) .

Multiply by 18:

18 \int_{1}^{2} f (x) d x = 2 (5 \ln 2 - 3) = 10 \ln 2 - 6.

Thus the answer is option C.

From NTA/IIT papers: JEE Main 2023 · 6 Apr Shift 1

Question 4 hard

The greatest integer less than or equal to

\int_{1}^{2} \log_{2} (x^{3} + 1) d x + \int_{1}^{\log_{2} 9} (2^{x} - 1)^{1 / 3} d x

is _____.

Answer: 5

Explanation. We start with the expression

I = \int_{1}^{2} \log_{2} (x^{3} + 1) d x + \int_{1}^{\log_{2} 9} (2^{x} - 1)^{1 / 3} d x .

**Step 1:** Change the base of the first logarithm:

\log_{2} (x^{3} + 1) = \frac{\ln (x^{3} + 1)}{\ln 2} .

**Step 2:** In the second integral, substitute

t = 2^{x} - 1 ⟹ d t = 2^{x} \ln 2 d x = (t + 1) \ln 2 d x .

When

x = 1

t = 1

; when

x = \log_{2} 9

t = 8

. Thus

\int_{1}^{\log_{2} 9} (2^{x} - 1)^{1 / 3} d x = \int_{1}^{8} t^{1 / 3} \frac{d t}{(t + 1) \ln 2} .

**Step 3:** Now consider the substitution in the first integral: let

u = x^{3} + 1 ⟹ d u = 3 x^{2} d x .

But this doesn’t directly match the second integral. Instead, notice the symmetry: Let

x = \log_{2} (y + 1)

in the first integral? That gives

d x = \frac{d y}{(y + 1) \ln 2}, x = 1 ⟹ y = 1, x = 2 ⟹ y = 3.

\int_{1}^{2} \log_{2} (x^{3} + 1) d x = \int_{1}^{3} \log_{2} ((\log_{2} (y + 1))^{3} + 1) \frac{d y}{(y + 1) \ln 2} .

This is messy, so instead use the known identity: For

f (x) = \log_{2} (x^{3} + 1)

, its inverse is

f^{- 1} (y) = (2^{y} - 1)^{1 / 3}

. Then by the inverse function property:

\int_{a}^{b} f (x) d x + \int_{f (a)}^{f (b)} f^{- 1} (y) d y = b f (b) - a f (a) .

**Step 4:** Here

a = 1

b = 2

. Compute:

f (1) = \log_{2} (1^{3} + 1) = \log_{2} 2 = 1, f (2) = \log_{2} (8 + 1) = \log_{2} 9.

\int_{1}^{2} \log_{2} (x^{3} + 1) d x + \int_{1}^{\log_{2} 9} (2^{y} - 1)^{1 / 3} d y = 2 \cdot \log_{2} 9 - 1 \cdot 1.

**Step 5:** Therefore

I = 2 \log_{2} 9 - 1 = 2 \log_{2} 9 - 1.

Now

\log_{2} 9 \approx 3.1699

, so

I \approx 2 (3.1699) - 1 = 6.3398 - 1 = 5.3398.

**Step 6:** The greatest integer less than or equal to

5.3398

5 .

From NTA/IIT papers: JEE Advanced 2022

Frequently asked questions

How much of JEE Main comes from Integral Calculus?: JEEGenie estimates about 8% of JEE Main marks come from Integral Calculus, based on recent papers. The share varies from year to year.
Which formulas matter most in Integral Calculus?: The Integral Calculus sheet on this page has 23 formula entries, starting with Indefinite Integral, Definite Integral, Power Rule for Integration, Integration of Trigonometric Functions, Integration of Exponential and Logarithmic Functions. Each one comes with when to use it and the trap to avoid.
What topics does Integral Calculus cover for JEE?: Integral Calculus covers Indefinite Integration, Methods of Integration, Definite Integrals, Properties of Definite Integrals, Area Under Curves, Reduction Formulae.
How should I revise Integral Calculus before the exam?: Learn each formula together with its condition of use, read the trap listed beside it, then practise questions and review every mistake. The practice link on this page opens Integral Calculus questions on JEEGenie.
How many previous-year questions are there on Integral Calculus?: JEEGenie has 1010 previous-year questions on Integral Calculus, and this page shows a few with worked explanations.
Is practice on Integral Calculus free on JEEGenie?: Sign-up is free and includes a 7-day Pro trial. After the trial, free accounts keep a limited daily quota.