# Integral Formula

Integral formulas on different functions are mentioned here. Apart from the formulas for integration, a brief introduction to integration, integral formulas classification and a few sample questions are also given here.

## Integration Definition

Integration can be considered as the reverse process as Differentiation or can be called Inverse Differentiation. Integration is the process of finding a function with its derivative.

## Integral Formulas Classification

Integration is the basic operation of Integral Calculus. Integral formulas are classified based on following funtions,

• Integral formulas based on Rational functions
• Integral formulas based on Irrational functions
• Integral formulas based on Trigonometric functions
• Integral formulas based on Inverse trigonometric functions
• Integral formulas based on Hyperbolic functions
• Integral formulas based on Inverse hyperbolic functions
• Integral formulas based on Exponential functions
• Integral formulas based on Logarithmic functions
• Integral formulas based on Gaussian functions

## Basic Integration Formulas List

Below is the list of Integral Formulas based on the above functions:

• $\large \int 1 \; dx = x + C$

• $\large \int a \; dx = ax + C$

• $\large \int x^{n} \; dx = \frac{x^{n+1}}{n+1} + C; \; n \neq -1$

• $\large \int \sin x \; dx = – \cos x + C$

• $\large \int \cos x \; dx = \sin x + C$

• $\large \int \sec ^{2} x \; dx = \tan x + C$

• $\large \int \csc ^{2} x \; dx = -\cot x + C$

• $\large \int \sec x (\tan x) \; dx = \sec x + C$

• $\large \int \csc x (\cot x) \; dx = – \csc x + C$

• $\large \int \frac{1}{x} \; dx = \ln |x| + C$

• $\large \int e^{x} \; dx = e^{x} + C$

• $\large \int a^{x} \; dx = \frac{a^{x}}{\ln a} + C; \; a> 0, a\neq 1$

• $\large \int \frac{1}{\sqrt{1-x^{2}}} \; dx = \sin^{-1} x + C$

• $\large \int \frac{1}{\sqrt{1+x^{2}}} \; dx = \tan^{-1} x + C$

• $\large \int \frac{1}{|x|\sqrt{x^{2}-1}} \; dx = \sec^{-1} x + C$

• $\large \int \sin^{n} (x) dx = \frac{-1}{n} \sin^{n-1} (x) \cos (x) +\frac{n-1}{n} \int \sin^{n-2} (x) dx$

• $\large \int \cos^{n} (x) dx = \frac{1}{n} \cos^{n-1} (x) \sin (x) +\frac{n-1}{n} \int \cos^{n-2} (x) dx$

• $\large \int \tan^{n} (x) dx = \frac{1}{n-1} \tan^{n-1} (x) +\int \tan^{n-2} (x) dx$

• $\large \int \sec^{n} (x) dx = \frac{1}{n-1} \sec^{n-2} (x) \tan (x) + \frac{n-2}{n-1} \int \sec^{n-2} (x) dx$

• $\large \int \csc^{n} (x) dx = \frac{-1}{n-1} \csc^{n-2} (x) \cot (x) + \frac{n-2}{n-1} \int \csc^{n-2} (x) dx$

## Question Based on Integration Formulas

• Question 1: Calculate $\int \,5x^{4}\,dx$
• Question 2: Find $\int x\sqrt{1+2x}\;dx$
• Question 2: $\int \frac{1}{x^{2}+6x+25}\,dx$
 More topics in Integral Formulas Integration by Parts Formula Definite Integral Formula Area under the Curve Formula U Substitution Formula Integration by Substitution Formula

#### Practise This Question

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