Math · Calculus

Integrals formulas for JEE

Every Integrals formula you need for JEE, grouped by concept.

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47 formulas4 concepts
01Indefinite Integration & Methods02Integration of Standard Types03Definite Integral Basics04Properties of Definite Integrals
01

Indefinite Integration & Methods

21 formulas

Antiderivative Definition

∫f(x) dx=F(x)+C\int f(x) \, dx = F(x) + C

The formula that gives all antiderivatives of a function, representing a family of parallel curves.

applies whenF′(x)=f(x)F'(x) = f(x)
indefiniteantiderivative

Integration by Parts

∫uv dx=u∫v dx−∫(u′∫v dx)dx\int u v \, dx = u \int v \, dx - \int \left( u' \int v \, dx \right) dx

ILATE rule is used to select the first function u.

applies whenu and v must be differentiable functions of x.
indefinitemethodsby_parts

Integral of a Constant

∫dx=x+C\int dx = x + C

Integration of the constant 1.

indefiniteconstant

Integral of Cosine

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

Standard integral of the cosine function.

indefinitetrigonometric

Integral of Cotangent

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

Derived via substitution.

indefinitetrigonometric

Integral of Cosecant

∫csc⁡x dx=log⁡∣csc⁡x−cot⁡x∣+C\int \csc x \, dx = \log|\csc x - \cot x| + C

Derived via substitution. Alternatively log⁡∣tan⁡(x2)∣+C\log|\tan(\frac{x}{2})| + C.

indefinitetrigonometric

Integral of Cosecant Squared

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

Standard integral of cosecant squared.

indefinitetrigonometric

Integral of Cosecant cotangent

∫csc⁡xcot⁡x dx=−csc⁡x+C\int \csc x \cot x \, dx = -\csc x + C

Standard integral of cosecant times cotangent.

indefinitetrigonometric

Integral of Exponential Function

∫ex dx=ex+C\int e^x \, dx = e^x + C

Standard integral of the natural exponential function.

indefiniteexponential

Integral of Exponential (Base a)

∫ax dx=axlog⁡a+C\int a^x \, dx = \frac{a^x}{\log a} + C

Standard integral for a general base exponential.

applies whena>0,a≠1a > 0, a \neq 1
indefiniteexponential

Classic e^x Pattern

∫ex[f(x)+f′(x)]dx=exf(x)+C\int e^x \left[ f(x) + f'(x) \right] dx = e^x f(x) + C

Derived by applying integration by parts on the e^x f(x) term.

indefinitemethodsby_parts

Integral resulting in Inverse Sine

∫dx1−x2=sin⁡−1x+C\int \frac{dx}{\sqrt{1-x^2}} = \sin^{-1} x + C

Also equals −cos⁡−1x+C-\cos^{-1} x + C.

applies when∣x∣<1|x| < 1
indefiniteinverse_trig

Integral resulting in Inverse Tangent

∫dx1+x2=tan⁡−1x+C\int \frac{dx}{1+x^2} = \tan^{-1} x + C

Also equals −cot⁡−1x+C-\cot^{-1} x + C.

indefiniteinverse_trig

Partial Fraction Decomposition (Linear)

px+q(x−a)(x−b)=Ax−a+Bx−b\frac{px+q}{(x-a)(x-b)} = \frac{A}{x-a} + \frac{B}{x-b}

Decomposition for non-repeated linear factors.

applies whena≠ba \neq b
indefinitemethodspartial_fractions

Power Rule for Integration

∫xn dx=xn+1n+1+C\int x^n \, dx = \frac{x^{n+1}}{n+1} + C

Integration of a polynomial base to a constant power.

applies whenn≠−1n \neq -1
indefinitepower_rule

Integral of Reciprocal

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

Integral of 1/x yielding natural logarithm.

applies whenx≠0x \neq 0
indefinitelogarithmic

Integral of Secant

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

Derived via substitution. Alternatively log⁡∣tan⁡(π4+x2)∣+C\log|\tan(\frac{\pi}{4} + \frac{x}{2})| + C.

indefinitetrigonometric

Integral of Secant Squared

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

Standard integral of secant squared.

indefinitetrigonometric

Integral of Secant tangent

∫sec⁡xtan⁡x dx=sec⁡x+C\int \sec x \tan x \, dx = \sec x + C

Standard integral of secant times tangent.

indefinitetrigonometric

Integral of Sine

∫sin⁡x dx=−cos⁡x+C\int \sin x \, dx = -\cos x + C

Standard integral of the sine function.

indefinitetrigonometric

Integral of Tangent

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

Derived via substitution.

indefinitetrigonometric
02

Integration of Standard Types

9 formulas

Integral of 1/sqrt(a^2 - x^2)

∫dxa2−x2=sin⁡−1(xa)+C\int \frac{dx}{\sqrt{a^2 - x^2}} = \sin^{-1} \left( \frac{x}{a} \right) + C

Standard irrational function integral.

applies when∣x∣<∣a∣|x| < |a|
indefinitestandard_form

Integral of 1/(x^2 + a^2)

∫dxx2+a2=1atan⁡−1(xa)+C\int \frac{dx}{x^2 + a^2} = \frac{1}{a} \tan^{-1} \left( \frac{x}{a} \right) + C

Standard rational function integral.

applies whena≠0a \neq 0
indefinitestandard_form

Integral of 1/sqrt(x^2 - a^2)

∫dxx2−a2=log⁡∣x+x2−a2∣+C\int \frac{dx}{\sqrt{x^2 - a^2}} = \log \left| x + \sqrt{x^2 - a^2} \right| + C

Standard irrational function integral.

applies when∣x∣>∣a∣|x| > |a|
indefinitestandard_form

Integral of 1/sqrt(x^2 + a^2)

∫dxx2+a2=log⁡∣x+x2+a2∣+C\int \frac{dx}{\sqrt{x^2 + a^2}} = \log \left| x + \sqrt{x^2 + a^2} \right| + C

Standard irrational function integral.

indefinitestandard_form

Integral of 1/(a^2 - x^2)

∫dxa2−x2=12alog⁡∣a+xa−x∣+C\int \frac{dx}{a^2 - x^2} = \frac{1}{2a} \log \left| \frac{a+x}{a-x} \right| + C

Standard rational function integral.

applies whenx≠±ax \neq \pm a
indefinitestandard_form

Integral of 1/(x^2 - a^2)

∫dxx2−a2=12alog⁡∣x−ax+a∣+C\int \frac{dx}{x^2 - a^2} = \frac{1}{2a} \log \left| \frac{x-a}{x+a} \right| + C

Standard rational function integral.

applies whenx≠±ax \neq \pm a
indefinitestandard_form

Integral of sqrt(a^2 - x^2)

∫a2−x2 dx=x2a2−x2+a22sin⁡−1(xa)+C\int \sqrt{a^2 - x^2} \, dx = \frac{x}{2} \sqrt{a^2 - x^2} + \frac{a^2}{2} \sin^{-1} \left( \frac{x}{a} \right) + C

Integral evaluated using integration by parts.

applies when∣x∣≤∣a∣|x| \leq |a|
indefinitestandard_form

Integral of sqrt(x^2 - a^2)

∫x2−a2 dx=x2x2−a2−a22log⁡∣x+x2−a2∣+C\int \sqrt{x^2 - a^2} \, dx = \frac{x}{2} \sqrt{x^2 - a^2} - \frac{a^2}{2} \log \left| x + \sqrt{x^2 - a^2} \right| + C

Integral evaluated using integration by parts.

applies when∣x∣≥∣a∣|x| \geq |a|
indefinitestandard_form

Integral of sqrt(x^2 + a^2)

∫x2+a2 dx=x2x2+a2+a22log⁡∣x+x2+a2∣+C\int \sqrt{x^2 + a^2} \, dx = \frac{x}{2} \sqrt{x^2 + a^2} + \frac{a^2}{2} \log \left| x + \sqrt{x^2 + a^2} \right| + C

Integral evaluated using integration by parts.

indefinitestandard_form
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03

Definite Integral Basics

4 formulas

Second Fundamental Theorem of Calculus

∫abf(x) dx=F(b)−F(a)\int_a^b f(x) \, dx = F(b) - F(a)

Evaluates a definite integral via the antiderivative FF.

applies whenff is continuous on [a,b][a, b], F′=fF' = f
definitefundamental_theorem

First Fundamental Theorem of Calculus

A′(x)=ddx∫axf(t) dt=f(x)A'(x) = \frac{d}{dx} \int_a^x f(t) \, dt = f(x)

The derivative of the area function is the original integrand.

applies whenff is continuous on [a,b][a, b], x∈[a,b]x \in [a, b]
definitefundamental_theorem

Definite Integral as Limit of a Sum

lim⁡n→∞1n∑r=1nf(rn)=∫01f(x) dx\lim_{n \to \infty} \frac{1}{n} \sum_{r=1}^n f\left(\frac{r}{n}\right) = \int_0^1 f(x) \, dx

Converts infinite Riemann sums into integration.

definitelimit_sumjee-advanced

Newton-Leibniz Formula

ddx∫g(x)h(x)f(t) dt=f(h(x))h′(x)−f(g(x))g′(x)\frac{d}{dx} \int_{g(x)}^{h(x)} f(t) \, dt = f(h(x))h'(x) - f(g(x))g'(x)

Differentiation under the integral sign.

applies whenh(x) and g(x) are differentiable, f(t) is continuous.
definitedifferentiationjee-advanced
04

Properties of Definite Integrals

13 formulas

Standard King's Rule Identity

∫0π/2sin⁡nxsin⁡nx+cos⁡nx dx=π4\int_0^{\pi/2} \frac{\sin^n x}{\sin^n x + \cos^n x} \, dx = \frac{\pi}{4}

Hidden identity found across several worked examples/exercises. Also valid for 1/(1+tan^n x) and 1/(1+cot^n x).

applies whenn∈Rn \in \mathbb{R}
definitepropertieskings_rule

Dummy Variable Property (P0)

∫abf(x) dx=∫abf(t) dt\int_a^b f(x) \, dx = \int_a^b f(t) \, dt

The value of a definite integral is independent of the variable of integration.

definiteproperties

Estimation of Definite Integrals

m(b−a)≤∫abf(x) dx≤M(b−a)m(b-a) \leq \int_a^b f(x) \, dx \leq M(b-a)

Bounds the value of an integral without evaluating it.

applies whenm≤f(x)≤M for x∈[a,b]m \leq f(x) \leq M \text{ for } x \in [a, b]
definiteestimationjee-advanced

Even / Odd Property (P7)

∫−aaf(x) dx=2∫0af(x) dx if even, 0 if odd\int_{-a}^a f(x) \, dx = 2 \int_0^a f(x) \, dx \text{ if even, } 0 \text{ if odd}

Simplifies symmetric intervals. Even: f(−x)=f(x)f(-x)=f(x), Odd: f(−x)=−f(x)f(-x)=-f(x).

applies whenSymmetric limits.
definitepropertieseven_odd

Half-Limit Property (P5)

∫02af(x) dx=∫0af(x) dx+∫0af(2a−x) dx\int_0^{2a} f(x) \, dx = \int_0^a f(x) \, dx + \int_0^a f(2a-x) \, dx

Splits the range from 0 to 2a into two integrals from 0 to a.

definiteproperties

King's Rule Special Case (P4)

∫0af(x) dx=∫0af(a−x) dx\int_0^a f(x) \, dx = \int_0^a f(a-x) \, dx

The most frequently used property in JEE integration questions.

definitepropertieskings_rule

King's Rule (P3)

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

Extremely powerful property to simplify complex integrands.

definitepropertieskings_rule

Limit Reversal Property (P1)

∫abf(x) dx=−∫baf(x) dx\int_a^b f(x) \, dx = -\int_b^a f(x) \, dx

Reversing limits changes the sign of the integral.

definiteproperties

Integral of log(sin x)

∫0π/2log⁡(sin⁡x) dx=−π2log⁡2\int_0^{\pi/2} \log(\sin x) \, dx = -\frac{\pi}{2} \log 2

Common integral derived via Properties 4 and 6 (Exercise Example 34).

definitestandard_form

Periodicity in Definite Integrals

∫0nTf(x) dx=n∫0Tf(x) dx\int_0^{nT} f(x) \, dx = n \int_0^T f(x) \, dx

Simplifies limits for periodic functions.

applies whenf(x)f(x) is periodic with period TT, n∈Zn \in \mathbb{Z}
definitepropertiesperiodicjee-advanced

Queen's Rule (P6)

∫02af(x) dx=2∫0af(x) dx if f(2a−x)=f(x)\int_0^{2a} f(x) \, dx = 2 \int_0^a f(x) \, dx \text{ if } f(2a-x) = f(x)

Evaluates to 0 if f(2a−x)=−f(x)f(2a-x) = -f(x).

definiteproperties

Splitting Property (P2)

∫abf(x) dx=∫acf(x) dx+∫cbf(x) dx\int_a^b f(x) \, dx = \int_a^c f(x) \, dx + \int_c^b f(x) \, dx

Useful for piecewise functions and absolute values.

definiteproperties

Wallis' Reduction Formula

∫0π/2sin⁡nxcos⁡mx dx=((n−1)(n−3)...)((m−1)(m−3)...)(n+m)(n+m−2)...×K\int_0^{\pi/2} \sin^n x \cos^m x \, dx = \frac{((n-1)(n-3)...)((m-1)(m-3)...)}{(n+m)(n+m-2)...} \times K

Reduction formula for trig powers. K = \pi/2 if both n, m are even, else K = 1.

applies whenn, m \geq 0 are integers.
definitereduction_formulajee-advanced
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