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Integration Formulas

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eSaral Academic and Editorial Team

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Summary

Integration formulas are the standard antiderivative rules used to reverse differentiation — for example, $\int x^n,dx=\frac{x^{n+1}}{n+1}+C$ (for $n\neq -1$) and $\int e^x,dx=e^x+C$. This integration formula list covers algebraic, trigonometric, exponential, logarithmic, and inverse trigonometric integral formulas, which together form the base of the entire Integrals chapter in Class 12 Mathematics and JEE.

Integration Formulas: Complete Integral Formulas List

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Every integration formula in this list is obtained by reversing a known derivative — if $\frac{d}{dx}[F(x)]=f(x)$, then $\int f(x),dx=F(x)+C$. This is the foundation of the entire integration formulas list used throughout the Integrals chapter for Class 12 Mathematics, JEE, and NEET.

Integration Formulas – Free PDF Download

Integration is one of the most important topics in calculus, and having the right integration formulas memorized can make the difference between solving a problem in seconds or getting stuck. This page brings together the complete integration formula list covering algebraic functions, trigonometric functions, inverse trigonometric functions, integration by parts, and definite integral properties all in one place. Whether you're preparing for JEE Main, JEE Advanced, or your Class 12 boards, these are the exact formulas you'll need on exam day. To make revision even easier, we've also compiled everything into a clean, printable integral formulas PDF that you can download and keep offline.

📥 Download the Integration Formulas PDF — a free, printable two-page reference covering every formula on this page. Download Integration Formulas PDF (Free)

Basic Algebraic Integration Formulas

Function

Integral Formula

$x^n$

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

$\frac{1}{x}$

$\int \frac{1}{x},dx=\ln\lvert x\rvert+C$

$1$ (constant)

$\int 1,dx=x+C$

$a$ (constant)

$\int a,dx=ax+C$

$e^x$

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

$a^x$

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

What Is the Integral Formula List for Trigonometric Functions?

The integral formulas for trigonometric functions in this integration formula list are used constantly in JEE and board numericals involving periodic functions.

Trigonometric Integration Formulas

Function

Integral Formula

$\sin x$

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

$\cos x$

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

$\sec^2 x$

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

$\csc^2 x$

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

$\sec x\tan x$

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

$\csc x\cot x$

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

$\tan x$

$\int \tan x,dx=\ln\lvert\sec x\rvert+C$

$\cot x$

$\int \cot x,dx=\ln\lvert\sin x\rvert+C$

$\sec x$

$\int \sec x,dx=\ln\lvert\sec x+\tan x\rvert+C$

$\csc x$

$\int \csc x,dx=\ln\lvert\csc x-\cot x\rvert+C$

What Are the Inverse Trigonometric Integral Formulas?

Inverse trigonometric integral formulas are one of the most tested parts of the integration formula list, since they involve recognisable square-root and quadratic patterns.

Inverse Trigonometric Integration Formulas

Standard Form

Integral Formula

$\frac{1}{\sqrt{a^2-x^2}}$

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

$\frac{1}{a^2+x^2}$

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

$\frac{1}{x\sqrt{x^2-a^2}}$

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

$\frac{1}{\sqrt{x^2-a^2}}$

$\int\frac{dx}{\sqrt{x^2-a^2}}=\ln\left\lvert x+\sqrt{x^2-a^2}\right\rvert+C$

$\frac{1}{\sqrt{x^2+a^2}}$

$\int\frac{dx}{\sqrt{x^2+a^2}}=\ln\left\lvert x+\sqrt{x^2+a^2}\right\rvert+C$

$\frac{1}{a^2-x^2}$

$\int\frac{dx}{a^2-x^2}=\frac{1}{2a}\ln\left\lvert\frac{a+x}{a-x}\right\rvert+C$

What Is the Formula for Integration by Parts?

Integration by parts is used when the integrand is a product of two functions, and it is one of the most important tools beyond the basic integral formulas list:

$\int u\cdot v,dx=u\int v,dx-\int\left(\frac{du}{dx}\int v,dx\right)dx$

The choice of $u$ and $v$ follows the ILATE rule (Inverse, Logarithmic, Algebraic, Trigonometric, Exponential), which decides the order of preference for selecting the first function $u$. This is one of the most frequently tested concepts in JEE Main Chapterwise PYQ

What Is the Difference Between Indefinite and Definite Integration Formulas?

While the integration formulas listed above give indefinite integrals (a family of functions plus $C$), definite integration formulas evaluate this antiderivative between two limits:

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

Key Definite Integral Properties

Property

Formula

Reversal of limits

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

Splitting the interval

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

King's property

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

Even function

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

Odd function

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

These properties — especially King's Property and the even/odd function shortcuts — appear repeatedly in JEE Advanced Chapterwise PYQ, often disguised inside trigonometric or rational integrands. 

Solved Examples Using the Integration Formula List

Applying the integration formulas list, once the pattern is recognised, follows a direct substitution into the matching standard result:

Example 1: Find $\int x^3,dx$

$\int x^3,dx=\frac{x^4}{4}+C$

Example 2: Find $\int\frac{dx}{\sqrt{4-x^2}}$

Here $a=2$, so using the inverse trig integral formula:

$\int\frac{dx}{\sqrt{4-x^2}}=\sin^{-1}\left(\frac{x}{2}\right)+C$

Example 3: Find $\int x,e^x,dx$

$\int x e^x,dx$ using integration by parts.

Taking $u=x$ and $dv=e^x,dx$:

$\int x e^x,dx=xe^x-\int e^x,dx=xe^x-e^x+C$

Example 4: Evaluate $\int_0^\pi\sin x,dx$

$\int_{0}^{\pi} \sin x\,dx = \left[-\cos x\right]_{0}^{\pi} = -\cos\pi + \cos 0 = 1 + 1 = 2$

Once these patterns feel familiar, test your recall speed with eSaral's JEE Test Series, which includes timed Integrals-focused mock sections.

Master indefinite integration in detail: Indefinite Integral Notes for Class 12 & IIT JEE →
Continue with definite integration and properties: Definite Integral Notes for Class 12 & IIT JEE →
Revise the complete chapter: Integration Class 12 Notes →
Apply these formulas to area problems: Application of Integrals Class 12 Notes →

Struggling to recall the right formula under exam pressure? eSaral's Class 12 Mathematics course, taught by IIT Bombay faculty, covers the complete Integrals chapter with derivation-first teaching and a 5-layer doubt-solving system. Explore Class 12 Maths Courses on eSaral →"

Frequently Asked Questions

What are integration formulas?

Integration formulas are standard antiderivative rules, such as $\int x^n,dx=\frac{x^{n+1}}{n+1}+C$, used to reverse differentiation and find the original function from its derivative.

What is the integral formula for $e^x$?

The integral formula for $e^x$ is $\int e^x,dx=e^x+C$, since $e^x$ is its own derivative and its own antiderivative.

What are the basic integral formulas for power functions?

The basic integral formula for power functions is $\int x^n,dx=\frac{x^{n+1}}{n+1}+C$ for $n\neq -1$, and $\int\frac{1}{x},dx=\ln\lvert x\rvert+C$ for the special case $n=-1$.

What is the complete integration formula list needed for JEE?

The integration formula list for JEE covers about 25–30 standard results spanning algebraic, exponential, logarithmic, trigonometric, and inverse trigonometric functions, along with integration by parts and standard substitution forms.

How many integration formulas should I memorise for Class 12 boards?

For Class 12 boards, roughly 20–25 standard integration formulas are sufficient, covering algebraic, trigonometric, exponential, logarithmic, and basic inverse trigonometric forms, along with the integration by parts formula.

What is the King's property in definite integration formulas?

The King's property states that $\int_a^b f(x),dx=\int_a^b f(a+b-x),dx$, and it is one of the most frequently used definite integral formulas in JEE for simplifying trigonometric fractions.

What is the difference between indefinite and definite integral formulas?

Indefinite integral formulas give a family of antiderivatives with a constant $C$, written as $\int f(x),dx=F(x)+C$, while definite integral formulas evaluate this result between two limits, written as $\int_a^b f(x),dx=F(b)-F(a)$.

What is the formula for integration by parts?

The formula for integration by parts is $\int u\cdot v,dx=u\int v,dx-\int\left(\frac{du}{dx}\int v,dx\right)dx$, used when integrating a product of two functions, with the first function chosen using the ILATE rule.

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