Integration formulas -

Introduction

Integration can feel tricky at first, but it becomes much easier once you remember the right formulas. Whether you’re preparing for CBSE Class 12 board exams, competitive exams like JEE, CUET, or just trying to strengthen your basics, having a clear list of integration formulas is essential.

Think of these formulas as your shortcuts — they save time, increase efficiency, and help you solve complex problems. In this formula sheet, we’ve put together all the important integration formulas you’ll need, starting from the simplest ones to slightly advanced standard forms.

These formulas cover:

Basic algebraic functions like polynomials and rational expressions.
Trigonometric integrals such as sine, cosine, and secant.
Exponential and logarithmic functions that frequently appear in board exam questions.
Special standard forms that are often used in NCERT and competitive exam problems.

By the end of this post, you’ll:

Have a complete integration formula sheet ready to refer to anytime.
Understand when and how to apply each formula.
Be ready to tackle NCERT exercises, CBSE board questions, and competitive exams with confidence.

Integration formulas for exponential functions

$\int x^n \, dx = \frac{1}{n+1}x^{n+1} + C$ For n≠−1

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

$\int a^x \, dx = \frac{a^x}{\ln(a)} + C$ For a≠1

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

Integration formulas for Trigonometric Functions

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

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

\int \tan(x) \, dx = \ln|\sec(x)| + C

\int \cosec(x) \, dx = \ln|\cosec(x) - \cot(x)| + C

\int \sec(x) \, dx = \ln|\sec(x) + \tan(x)| + C

\int \cot(x) \, dx = \ln|\sin(x)| + C

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

\int \frac{1}{\sqrt{1 - x^2}} \, dx = -\cos^{-1}(x) + C

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

\int \frac{1}{1 + x^2} \, dx = -\cot^{-1}(x) + C

\int \frac{1}{x\sqrt{x^2 - 1}} \, dx = \sec^{-1}(x) + C

\int \frac{1}{x\sqrt{x^2 - 1}} \, dx = -\cosec^{-1}(x) + C

Integral formulas for some special functions.

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

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

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

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

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

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

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

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

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

For properties of definite integration go through the following link: Properties of definite integration

For integration Problems to practice from NCERT go through the following link: Solutions to class 12 NCERT problems

For complete list of class 12 Maths important formulas go through the following link: Class 12 Maths Important Formulae

Introduction

Integration formulas for exponential functions

Integration formulas for Trigonometric Functions

Integral formulas for some special functions.

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