Mathematics · JEE

Integral Calculus Formula Sheet for JEE

95+ JEE formulas in this unit

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The Integral Calculus JEE formula sheet lists 95+ important formulas for JEE Main and Advanced, including essential identities from Integral as an anti-derivative, Fundamental integrals involving algebraic, trigonometric, exponential and logarithmic functions, Integration by substitution, by parts and by partial fractions, Integration using trigonometric identities, and more. Revise essential formulas first, then practise MCQs on Goodmarks.

Download-free JEE mathematics formula revision for Integral Calculus. This unit-wise formula list covers 95+ exam-relevant results across Integral as an anti-derivative, Fundamental integrals involving algebraic, trigonometric, exponential and logarithmic functions, Integration by substitution, by parts and by partial fractions, Integration using trigonometric identities, and more, organised by subtopic for quick last-minute revision.

JEE Formula Sheet

95 formulas across 8 subtopics — organised for JEE Main & Advanced revision

Practise MCQs for this unit

Essential: 37Important: 40Supplementary: 18

Integral as an anti-derivative

Indefinite integral as antiderivativeEssential

\int f(x)\,dx = F(x)+C \quad \text{where } F'(x)=f(x)

Variables: $F(x)$ is any antiderivative of $f(x)$ ; $C$ is the constant of integration.
Conditions: Applicable where $f$ admits an antiderivative on the interval considered.
Where used in JEE: Basic indefinite integration; solving antiderivative-based questions.

Linearity of indefinite integrationEssential

\int \big(af(x)\pm bg(x)\big)\,dx = a\int f(x)\,dx \pm b\int g(x)\,dx

Variables: $a,b$ are constants.
Conditions: For integrable functions $f,g$ .
Where used in JEE: Breaking integrals into simpler parts.

Fundamental integrals involving algebraic, trigonometric, exponential and logarithmic functions

Power rule for integrationEssential

\int x^n\,dx = \frac{x^{n+1}}{n+1}+C, \qquad n\neq -1

Variables: $n$ is a real number.
Conditions: $n\neq -1$ .
Where used in JEE: Most algebraic indefinite and definite integrals.

Integral of reciprocal functionEssential

\int \frac{1}{x}\,dx = \log|x|+C

Conditions: $x\neq 0$ .
Where used in JEE: Logarithmic integrals; substitution; partial fractions.

Integral of constantImportant

\int k\,dx = kx+C

Variables: $k$ is a constant.
Where used in JEE: Basic antiderivative evaluation.

Integral of exponential with base eEssential

\int e^x\,dx=e^x+C

Where used in JEE: Exponential function integrals.

Integral of general exponentialImportant

\int a^x\,dx=\frac{a^x}{\log a}+C

Variables: $a$ is the base.
Conditions: $a>0,\ a\neq 1$ .
Where used in JEE: Exponential integrals with non-natural base.

Integral of logarithmImportant

\int \log x\,dx=x\log x-x+C

Conditions: $x>0$ .
Where used in JEE: By parts; logarithmic integrals.

Integral of sineEssential

\int \sin x\,dx=-\cos x+C

Where used in JEE: Basic trigonometric integration.

Integral of cosineEssential

\int \cos x\,dx=\sin x+C

Where used in JEE: Basic trigonometric integration.

Integral of secant squareEssential

\int \sec^2 x\,dx=\tan x+C

Conditions: Defined where $\cos x\neq 0$ .
Where used in JEE: Standard trigonometric integrals.

Integral of cosecant squareEssential

\int \csc^2 x\,dx=-\cot x+C

Conditions: Defined where $\sin x\neq 0$ .
Where used in JEE: Standard trigonometric integrals.

Integral of secant tangentEssential

\int \sec x\tan x\,dx=\sec x+C

Conditions: Defined where $\cos x\neq 0$ .
Where used in JEE: Standard trigonometric integrals; substitution.

Integral of cosecant cotangentEssential

\int \csc x\cot x\,dx=-\csc x+C

Conditions: Defined where $\sin x\neq 0$ .
Where used in JEE: Standard trigonometric integrals; substitution.

Integral of tangentImportant

\int \tan x\,dx=\log|\sec x|+C=-\log|\cos x|+C

Conditions: Defined where $\cos x\neq 0$ .
Where used in JEE: Trigonometric simplification and standard forms.

Integral of cotangentImportant

\int \cot x\,dx=\log|\sin x|+C

Conditions: Defined where $\sin x\neq 0$ .
Where used in JEE: Trigonometric simplification and standard forms.

Integral of secantImportant

\int \sec x\,dx=\log|\sec x+\tan x|+C

Conditions: Defined where $\cos x\neq 0$ .
Where used in JEE: Important standard trigonometric result.

Integral of cosecantImportant

\int \csc x\,dx=\log|\csc x-\cot x|+C

Conditions: Defined where $\sin x\neq 0$ .
Where used in JEE: Important standard trigonometric result.

Integral of inverse square root formEssential

\int \frac{1}{\sqrt{a^2-x^2}}\,dx=\sin^{-1}\!\left(\frac{x}{a}\right)+C

Variables: $a$ is a nonzero constant.
Conditions: Usually taken for $a>0$ and $|x|<a$ .
Where used in JEE: Standard algebraic forms; substitution to inverse trigonometric result.

Integral of reciprocal quadratic root formEssential

\int \frac{1}{a^2+x^2}\,dx=\frac{1}{a}\tan^{-1}\!\left(\frac{x}{a}\right)+C

Variables: $a$ is a nonzero constant.
Conditions: Usually $a>0$ .
Where used in JEE: Standard inverse trigonometric form; partial fractions and substitution.

Integral of reciprocal square-root difference formImportant

\int \frac{1}{\sqrt{x^2-a^2}}\,dx=\log\left|x+\sqrt{x^2-a^2}\right|+C

Variables: $a$ is a nonzero constant.
Conditions: Valid on domains where $|x|>|a|$ .
Where used in JEE: Standard algebraic irrational integrals.

Integral of reciprocal square-root sum formImportant

\int \frac{1}{\sqrt{x^2+a^2}}\,dx=\log\left|x+\sqrt{x^2+a^2}\right|+C

Variables: $a$ is a constant.
Conditions: Usually $a>0$ .
Where used in JEE: Standard algebraic irrational integrals.

Integral of reciprocal linear-square-root formImportant

\int \frac{1}{x^2-a^2}\,dx=\frac{1}{2a}\log\left|\frac{x-a}{x+a}\right|+C

Variables: $a\neq 0$ .
Conditions: $x\neq \pm a$ .
Where used in JEE: Partial fractions and standard rational forms.

Integration by substitution, by parts and by partial fractions

Substitution rule for indefinite integralsEssential

\int f(g(x))g'(x)\,dx=\int f(t)\,dt\quad \text{with } t=g(x)

Variables: $t$ is the substituted variable.
Conditions: $g$ differentiable and substitution valid on the domain.
Where used in JEE: u-substitution in composite integrands.

Linear substitution formsImportant

\int f(ax+b)\,dx=\frac{1}{a}\int f(t)\,dt \quad (t=ax+b)

Variables: $a,b$ are constants.
Conditions: $a\neq 0$ .
Where used in JEE: Quick evaluation of linear-argument integrals.

Integration by partsEssential

\int u\,dv=uv-\int v\,du

Variables: $u=u(x)$ , $v=v(x)$ , $du=u'(x)dx$ , $dv=v'(x)dx$ .
Conditions: Functions should be differentiable/integrable as needed.
Where used in JEE: Products like algebraic-trigonometric, algebraic-exponential, logarithmic integrals.

LIATE priority for choosing first function in by-partsSupplementary

L>I>A>T>E

Variables: $L$ : logarithmic, $I$ : inverse trigonometric, $A$ : algebraic, $T$ : trigonometric, $E$ : exponential.
Conditions: Heuristic, not a theorem.
Where used in JEE: Selecting $u$ in integration by parts.

Partial fractions for distinct linear factorsEssential

\frac{P(x)}{(x-a)(x-b)}=\frac{A}{x-a}+\frac{B}{x-b}

Variables: $P(x)$ is a polynomial of lower degree than denominator; $A,B$ constants.
Conditions: $a\neq b$ ; proper rational function.
Where used in JEE: Integration of rational functions.

Partial fractions for repeated linear factorImportant

\frac{P(x)}{(x-a)^n}=\frac{A_1}{x-a}+\frac{A_2}{(x-a)^2}+\cdots+\frac{A_n}{(x-a)^n}

Variables: $A_1,\dots,A_n$ are constants.
Conditions: Proper rational function; $n\in\mathbb{N}$ .
Where used in JEE: Rational function integration with repeated roots.

Partial fractions for irreducible quadratic factorImportant

\frac{P(x)}{x^2+px+q}=\frac{Ax+B}{x^2+px+q}

Variables: $A,B$ are constants.
Conditions: For proper rational decomposition when $x^2+px+q$ is irreducible over reals.
Where used in JEE: Rational functions leading to logarithm and arctangent forms.

Partial fractions for repeated irreducible quadratic factorSupplementary

\frac{P(x)}{(x^2+px+q)^n}=\frac{A_1x+B_1}{x^2+px+q}+\frac{A_2x+B_2}{(x^2+px+q)^2}+\cdots+\frac{A_nx+B_n}{(x^2+px+q)^n}

Variables: $A_i,B_i$ are constants.
Conditions: Proper rational function; quadratic irreducible over reals.
Where used in JEE: Advanced partial fraction decomposition.

Reduction of improper rational functionImportant

\frac{P(x)}{Q(x)}=S(x)+\frac{R(x)}{Q(x)},\quad \deg R<\deg Q

Variables: $S(x)$ is quotient, $R(x)$ remainder.
Conditions: Use polynomial division when $\deg P\ge \deg Q$ .
Where used in JEE: Preparing rational functions for partial fractions.

Tangent half-angle substitution identitiesImportant

t=\tan\frac{x}{2},\quad \sin x=\frac{2t}{1+t^2},\quad \cos x=\frac{1-t^2}{1+t^2},\quad dx=\frac{2}{1+t^2}\,dt

Variables: $t$ is the new variable.
Conditions: Useful for rational functions of $\sin x$ and $\cos x$ .
Where used in JEE: Universal trigonometric substitution.

Integration using trigonometric identities

Pythagorean identitiesEssential

\sin^2 x+\cos^2 x=1,\quad 1+\tan^2 x=\sec^2 x,\quad 1+\cot^2 x=\csc^2 x

Where used in JEE: Simplifying trigonometric integrands.

Double-angle power reduction identitiesEssential

\sin^2 x=\frac{1-\cos 2x}{2},\qquad \cos^2 x=\frac{1+\cos 2x}{2}

Where used in JEE: Integrals involving powers of sine and cosine.

Product identity for sine-cosine squareImportant

\sin x\cos x=\frac{1}{2}\sin 2x

Where used in JEE: Simplifying products of sine and cosine.

Product-to-sum identitiesImportant

\sin A\sin B=\frac{1}{2}\big[\cos(A-B)-\cos(A+B)\big]

\cos A\cos B=\frac{1}{2}\big[\cos(A-B)+\cos(A+B)\big]

\sin A\cos B=\frac{1}{2}\big[\sin(A+B)+\sin(A-B)\big]

Variables: $A,B$ are angles/expressions.
Where used in JEE: Integration of trigonometric products.

Triple-angle based power identity for sine cubeSupplementary

\sin^3 x=\frac{3\sin x-\sin 3x}{4}

Where used in JEE: Integrals of odd powers of sine.

Triple-angle based power identity for cosine cubeSupplementary

\cos^3 x=\frac{3\cos x+\cos 3x}{4}

Where used in JEE: Integrals of odd powers of cosine.

Reduction of tangent square and cotangent squareImportant

\tan^2 x=\sec^2 x-1,\qquad \cot^2 x=\csc^2 x-1

Where used in JEE: Integrals of powers of tangent and cotangent.

Standard strategy for powers of sine and cosineSupplementary

\int \sin^m x\cos^n x\,dx

Variables: $m,n$ are nonnegative integers.
Conditions: If one power is odd, separate one factor and substitute; if both even, use half-angle identities.
Where used in JEE: Evaluation of trigonometric power integrals.

Standard strategy for powers of secant and tangentSupplementary

\int \sec^m x\tan^n x\,dx

Variables: $m,n$ are integers.
Conditions: If $n$ is odd, separate $\sec x\tan x$ ; if $m$ is even, use $\tan^2 x=\sec^2 x-1$ .
Where used in JEE: Evaluation of secant-tangent power integrals.

Evaluation of simple integrals of standard algebraic/trigonometric forms

Integral of reciprocal root-product formSupplementary

\int \frac{1}{\sqrt{(x-a)(b-x)}}\,dx=\sin^{-1}\!\left(\frac{2x-a-b}{b-a}\right)+C

Variables: $a,b$ are constants with $a<b$ .
Conditions: Typically $a<x<b$ .
Where used in JEE: Special standard algebraic form in JEE.

Integral of sine squareImportant

\int \sin^2 x\,dx=\frac{x}{2}-\frac{\sin 2x}{4}+C

Where used in JEE: Standard trigonometric power integral.

Integral of cosine squareImportant

\int \cos^2 x\,dx=\frac{x}{2}+\frac{\sin 2x}{4}+C

Where used in JEE: Standard trigonometric power integral.

Integral of sine cubeSupplementary

\int \sin^3 x\,dx=-\cos x+\frac{\cos^3 x}{3}+C

Where used in JEE: Odd power trigonometric integral.

Integral of cosine cubeSupplementary

\int \cos^3 x\,dx=\sin x-\frac{\sin^3 x}{3}+C

Where used in JEE: Odd power trigonometric integral.

Integral of secant cubeSupplementary

\int \sec^3 x\,dx=\frac{1}{2}\big(\sec x\tan x+\log|\sec x+\tan x|\big)+C

Conditions: Defined where $\cos x\neq 0$ .
Where used in JEE: Special standard trigonometric integral.

Integral of cosecant cubeSupplementary

\int \csc^3 x\,dx=\frac{1}{2}\big(-\csc x\cot x+\log|\csc x-\cot x|\big)+C

Conditions: Defined where $\sin x\neq 0$ .
Where used in JEE: Special standard trigonometric integral.

Integral of reciprocal square-root linear formImportant

\int \frac{1}{\sqrt{ax+b}}\,dx=\frac{2}{a}\sqrt{ax+b}+C

Variables: $a,b$ are constants.
Conditions: $a\neq 0$ ; radicand defined.
Where used in JEE: Simple substitution-based algebraic integral.

Integral of square-root linear formImportant

\int \sqrt{ax+b}\,dx=\frac{2(ax+b)^{3/2}}{3a}+C

Variables: $a,b$ are constants.
Conditions: $a\neq 0$ ; radicand defined.
Where used in JEE: Simple substitution-based algebraic integral.

Integral of reciprocal square-root quadratic difference formEssential

\int \frac{1}{\sqrt{a^2-x^2}}\,dx=\sin^{-1}\!\left(\frac{x}{a}\right)+C

Variables: $a>0$ .
Conditions: $|x|<a$ .
Where used in JEE: Inverse trigonometric standard form.

Integral of reciprocal square-root quadratic sum formImportant

\int \frac{1}{\sqrt{x^2+a^2}}\,dx=\log\left|x+\sqrt{x^2+a^2}\right|+C

Variables: $a>0$ .
Where used in JEE: Standard irrational quadratic form.

Integral of reciprocal square-root quadratic excess formImportant

\int \frac{1}{\sqrt{x^2-a^2}}\,dx=\log\left|x+\sqrt{x^2-a^2}\right|+C

Variables: $a>0$ .
Conditions: $|x|>a$ .
Where used in JEE: Standard irrational quadratic form.

Integral of reciprocal shifted quadraticImportant

\int \frac{1}{(x-a)^2+b^2}\,dx=\frac{1}{b}\tan^{-1}\!\left(\frac{x-a}{b}\right)+C

Variables: $a,b$ are constants.
Conditions: $b>0$ .
Where used in JEE: Completing square in rational integrals.

Integral of derivative over functionEssential

\int \frac{f'(x)}{f(x)}\,dx=\log|f(x)|+C

Variables: $f(x)$ differentiable.
Conditions: $f(x)\neq 0$ .
Where used in JEE: Substitution; logarithmic form detection.

Integral of derivative over quadratic sum formEssential

\int \frac{f'(x)}{a^2+f(x)^2}\,dx=\frac{1}{a}\tan^{-1}\!\left(\frac{f(x)}{a}\right)+C

Variables: $a\neq 0$ .
Conditions: Usually $a>0$ .
Where used in JEE: Substitution to inverse tangent form.

Integral of derivative over square-root difference formEssential

\int \frac{f'(x)}{\sqrt{a^2-f(x)^2}}\,dx=\sin^{-1}\!\left(\frac{f(x)}{a}\right)+C

Variables: $a\neq 0$ .
Conditions: Usually $a>0$ , $|f(x)|<a$ .
Where used in JEE: Substitution to inverse sine form.

The fundamental theorem of calculus, properties of definite integrals

First fundamental theorem of calculusEssential

\frac{d}{dx}\left(\int_a^x f(t)\,dt\right)=f(x)

Variables: $t$ is dummy variable; $a$ constant.
Conditions: $f$ continuous at $x$ .
Where used in JEE: Differentiation of integral-defined functions.

Generalized FTC with variable limitsEssential

\frac{d}{dx}\left(\int_{\phi(x)}^{\psi(x)} f(t)\,dt\right)=f(\psi(x))\psi'(x)-f(\phi(x))\phi'(x)

Variables: $\phi(x),\psi(x)$ are differentiable limit functions.
Conditions: $f$ continuous on relevant interval.
Where used in JEE: Differentiation under variable limits.

Newton-Leibniz formulaEssential

\int_a^b f(x)\,dx=F(b)-F(a)\quad \text{where } F'(x)=f(x)

Variables: $F$ is an antiderivative of $f$ .
Conditions: $f$ integrable on $[a,b]$ .
Where used in JEE: Evaluation of definite integrals.

Zero interval propertyImportant

\int_a^a f(x)\,dx=0

Where used in JEE: Basic definite integral properties.

Reversal of limitsEssential

\int_a^b f(x)\,dx=-\int_b^a f(x)\,dx

Where used in JEE: Changing order of integration limits.

Additivity over intervalsEssential

\int_a^b f(x)\,dx=\int_a^c f(x)\,dx+\int_c^b f(x)\,dx

Variables: $c$ lies between $a$ and $b$ .
Conditions: $f$ integrable on the interval.
Where used in JEE: Breaking definite integrals at convenient points.

Linearity of definite integrationEssential

\int_a^b \big(\alpha f(x)+\beta g(x)\big)\,dx=\alpha\int_a^b f(x)\,dx+\beta\int_a^b g(x)\,dx

Variables: $\alpha,\beta$ are constants.
Conditions: $f,g$ integrable on $[a,b]$ .
Where used in JEE: Splitting and combining definite integrals.

Comparison propertyImportant

f(x)\ge g(x)\ \forall x\in[a,b] \implies \int_a^b f(x)\,dx\ge \int_a^b g(x)\,dx

Conditions: Functions integrable on $[a,b]$ .
Where used in JEE: Bounding and sign of definite integrals.

Non-negativity propertyImportant

f(x)\ge 0\ \forall x\in[a,b] \implies \int_a^b f(x)\,dx\ge 0

Conditions: $f$ integrable on $[a,b]$ .
Where used in JEE: Area interpretation and sign analysis.

Absolute value inequalitySupplementary

\left|\int_a^b f(x)\,dx\right|\le \int_a^b |f(x)|\,dx

Conditions: $f$ integrable on $[a,b]$ .
Where used in JEE: Bounding definite integrals.

Property under substitution in definite integralsEssential

\int_a^b f(g(x))g'(x)\,dx=\int_{g(a)}^{g(b)} f(t)\,dt

Variables: $t=g(x)$ .
Conditions: Valid when substitution is differentiable and monotone or otherwise legitimate.
Where used in JEE: Definite integral evaluation by substitution.

Integration by parts for definite integralsImportant

\int_a^b u\,dv=\big[uv\big]_a^b-\int_a^b v\,du

Variables: $u,v$ are suitable functions.
Conditions: Required derivatives/integrals exist on $[a,b]$ .
Where used in JEE: Definite integrals of products.

Symmetry property with x to a+b-xEssential

\int_a^b f(x)\,dx=\int_a^b f(a+b-x)\,dx

Conditions: $f$ integrable on $[a,b]$ .
Where used in JEE: JEE symmetry-based definite integral simplification.

Pairing property using symmetryEssential

\int_a^b f(x)\,dx=\frac{1}{2}\int_a^b \big(f(x)+f(a+b-x)\big)\,dx

Conditions: $f$ integrable on $[a,b]$ .
Where used in JEE: Transforming hard integrands into simpler symmetric forms.

Odd function over symmetric intervalEssential

f(-x)=-f(x) \implies \int_{-a}^{a} f(x)\,dx=0

Conditions: $f$ integrable on $[-a,a]$ .
Where used in JEE: Symmetry-based evaluation.

Even function over symmetric intervalEssential

f(-x)=f(x) \implies \int_{-a}^{a} f(x)\,dx=2\int_0^{a} f(x)\,dx

Conditions: $f$ integrable on $[-a,a]$ .
Where used in JEE: Symmetry-based evaluation.

Trigonometric complementary property on [0,a]Essential

\int_0^a f(x)\,dx=\int_0^a f(a-x)\,dx

Conditions: $f$ integrable on $[0,a]$ .
Where used in JEE: Especially with $a=\frac{\pi}{2}$ or $\pi$ in trigonometric integrals.

Evaluation of definite integrals

Definite integral symmetry on [0,pi/2]Important

\int_0^{\pi/2} f(\sin x,\cos x)\,dx=\int_0^{\pi/2} f(\cos x,\sin x)\,dx

Variables: $f$ is an expression in two arguments.
Conditions: Integrable on $[0,\pi/2]$ .
Where used in JEE: Swapping sine and cosine in definite integrals.

Definite integral of powers of sine and cosine equalityImportant

\int_0^{\pi/2} \sin^m x\cos^n x\,dx=\int_0^{\pi/2} \sin^n x\cos^m x\,dx

Variables: $m,n$ are real numbers for which integral exists.
Where used in JEE: Symmetry in trigonometric definite integrals.

Integral of sine over [0,pi/2]Important

\int_0^{\pi/2} \sin x\,dx=1

Where used in JEE: Standard value.

Integral of cosine over [0,pi/2]Important

\int_0^{\pi/2} \cos x\,dx=1

Where used in JEE: Standard value.

Integral of sine square and cosine square over [0,pi/2]Important

\int_0^{\pi/2} \sin^2 x\,dx=\int_0^{\pi/2} \cos^2 x\,dx=\frac{\pi}{4}

Where used in JEE: Standard trigonometric definite values.

Integral of reciprocal linear over intervalSupplementary

\int_0^a \frac{1}{x+b}\,dx=\log\left|\frac{a+b}{b}\right|

Variables: $a,b$ are constants.
Conditions: No singularity on $[0,a]$ ; in particular $b\neq 0$ and $-b\notin[0,a]$ .
Where used in JEE: Basic definite logarithmic evaluation.

Standard Wallis-type reduction for sine powersImportant

I_n=\int_0^{\pi/2}\sin^n x\,dx=\frac{n-1}{n}I_{n-2}

Variables: $n\ge 2$ ; $I_n$ denotes the integral.
Conditions: Similarly valid for $\cos^n x$ . Bases: $I_0=\frac{\pi}{2},\ I_1=1$ .
Where used in JEE: Evaluation of trigonometric power definite integrals.

Wallis formula for even powersSupplementary

\int_0^{\pi/2}\sin^{2n}x\,dx=\int_0^{\pi/2}\cos^{2n}x\,dx=\frac{(2n-1)!!}{(2n)!!}\cdot\frac{\pi}{2}

Variables: $n\in\mathbb{N}$ ; $(2n-1)!!=1\cdot 3\cdot 5\cdots(2n-1)$ , $(2n)!!=2\cdot 4\cdot 6\cdots 2n$ .
Where used in JEE: Standard definite values for even powers.

Wallis formula for odd powersSupplementary

\int_0^{\pi/2}\sin^{2n+1}x\,dx=\int_0^{\pi/2}\cos^{2n+1}x\,dx=\frac{(2n)!!}{(2n+1)!!}

Variables: $n\in\mathbb{N}_0$ .
Where used in JEE: Standard definite values for odd powers.

Beta-type evaluation of mixed powersSupplementary

\int_0^{\pi/2}\sin^m x\cos^n x\,dx=\frac{(m-1)(m-3)\cdots}{(m+n)(m+n-2)\cdots}\times\begin{cases}\frac{\pi}{2},& m,n\text{ both even}\\1,& \text{otherwise in standard Wallis form}\end{cases}

Variables: $m,n$ are nonnegative integers.
Conditions: Interpret products until reaching 1 or 2 appropriately.
Where used in JEE: JEE evaluation of mixed power trigonometric integrals.

Determining areas of the regions bounded by simple curves in standard forms

Area under a curve above x-axisEssential

\text{Area}=\int_a^b y\,dx=\int_a^b f(x)\,dx

Variables: $y=f(x)$ .
Conditions: Applicable when $f(x)\ge 0$ on $[a,b]$ .
Where used in JEE: Area bounded by curve, x-axis, and vertical lines.

Area between curve and x-axis in generalEssential

\text{Area}=\int_a^b |f(x)|\,dx

Variables: $y=f(x)$ .
Conditions: Use when curve crosses the x-axis.
Where used in JEE: Unsigned area of regions with sign changes.

Area between two curves using xEssential

\text{Area}=\int_a^b \big(y_{\text{upper}}-y_{\text{lower}}\big)\,dx

Variables: $a,b$ are x-coordinates of intersection points or given boundaries.
Conditions: Upper curve should remain above lower curve on the interval, else split.
Where used in JEE: Region bounded by two curves in standard form $y=f(x)$.

Area between two curves using yImportant

\text{Area}=\int_c^d \big(x_{\text{right}}-x_{\text{left}}\big)\,dy

Variables: $c,d$ are y-limits.
Conditions: Right curve should remain to the right of left curve, else split.
Where used in JEE: When curves are easier as $x=g(y)$.

Area bounded by curve and y-axisImportant

\text{Area}=\int_c^d x\,dy

Variables: $x=g(y)$ .
Conditions: Applicable when $x\ge 0$ on $[c,d]$ .
Where used in JEE: Regions naturally expressed in terms of $y$.

Area using symmetryImportant

\text{Total Area}=2\times \text{Area of one symmetric part}\quad \text{or}\quad 4\times \text{Area of one quadrant part}

Conditions: Use only when the region is symmetric about an axis or the origin.
Where used in JEE: Circles, parabolas, ellipses, and symmetric bounded regions.

Area of region bounded by parabola and its latus-rectum/chord forms via integration setupSupplementary

\text{If } y^2=4ax,\ \text{then area between } x=\frac{y^2}{4a} \text{ and } x=h \text{ is } \int_{y_1}^{y_2}\left(h-\frac{y^2}{4a}\right)dy

Variables: $a,h$ are constants; $y_1,y_2$ are intersection ordinates.
Conditions: Choose limits from intersection points.
Where used in JEE: Standard parabola-bounded area questions.

Area of region bounded by circle and axis/chord forms via integration setupImportant

\text{If } x^2+y^2=a^2,\ \text{then upper semicircle } y=\sqrt{a^2-x^2},\ \text{and area }=\int (\text{top}-\text{bottom})\,dx

Variables: $a$ is radius.
Conditions: Split using symmetry when convenient.
Where used in JEE: Circle segment and semicircle area problems.

Area of full circle by integrationImportant

\text{Area of } x^2+y^2=a^2=4\int_0^a \sqrt{a^2-x^2}\,dx=\pi a^2

Variables: $a$ is radius.
Conditions: $a>0$ .
Where used in JEE: Standard application of definite integrals to area.

Area of ellipse by integrationSupplementary

\text{Area of } \frac{x^2}{a^2}+\frac{y^2}{b^2}=1 =4\int_0^a b\sqrt{1-\frac{x^2}{a^2}}\,dx=\pi ab

Variables: $a,b>0$ are semi-axes.
Where used in JEE: Standard bounded area result.

Frequently asked questions

What are the important Integral Calculus formulas for JEE?

This page lists 95+ JEE-relevant Integral Calculus formulas organised by subtopic. Start with essential formulas, then important identities before supplementary shortcuts.

Is this Integral Calculus formula sheet aligned with JEE Main?

Yes. Every formula is mapped to the JEE Main Mathematics syllabus for Integral Calculus, covering Integral as an anti-derivative, Fundamental integrals involving algebraic, trigonometric, exponential and logarithmic functions, Integration by substitution, by parts and by partial fractions, Integration using trigonometric identities, and more.

How should I revise the Integral Calculus formula sheet before JEE?

Revise essential formulas daily, important ones every 2–3 days, and supplementary results weekly. After each pass, solve 10–15 MCQs to test recall under exam conditions.

Where can I practise Integral Calculus MCQs after revising formulas?

Use the Online Practice or MCQs pages for the same unit on Goodmarks to convert formula recall into problem-solving speed.

Does this replace NCERT for Integral Calculus?

No — use this formula sheet for quick revision alongside NCERT and your coaching notes. Formulas here are a condensed reference, not a substitute for concept building.