# Integral rule of Reciprocal of Sum of One and Square of variable

## Formula

$\displaystyle \int{\dfrac{1}{1+x^2}\,\,}dx$ $\,=\,$ $\tan^{-1}{x}+c \,\,\,$ or $\,\,\, \arctan{(x)}+c$

### Introduction

When $x$ is considered to represent a variable, the sum of one and square of variable $x$ is written as $1+x^2$ mathematically. The inverse tangent function written as $\tan^{-1}{x}$ or $\arctan{(x)}$ in mathematics. The integral of the reciprocal of the expression $1+x^2$ is expressed in the following mathematical form.

$\displaystyle \int{\dfrac{1}{1+x^2}\,\,}dx$

The indefinite integral of the rational expression with respect to $x$ is equal to the tan inverse of $x$.

$(1)\,\,\,$ $\displaystyle \int{\dfrac{1}{1+x^2}\,\,}dx$ $\,=\,$ $\tan^{-1}{x}+c$

$(2)\,\,\,$ $\displaystyle \int{\dfrac{1}{1+x^2}\,\,}dx$ $\,=\,$ $\arctan{(x)}+c$

#### Alternative form

The integral law of reciprocal sum of one and square of variable can be expressed in terms of any variable.

$(1)\,\,\,$ $\displaystyle \int{\dfrac{1}{1+l^2}\,\,}dl$ $\,=\,$ $\tan^{-1}{l}+c$

$(2)\,\,\,$ $\displaystyle \int{\dfrac{1}{1+q^2}\,\,}dq$ $\,=\,$ $\arctan{(q)}+c$

$(3)\,\,\,$ $\displaystyle \int{\dfrac{1}{1+y^2}\,\,}dy$ $\,=\,$ $\tan^{-1}{y}+c$

##### Proof

Learn how to prove the integration formula for the multiplicative inverse of one plus variable squared in integral calculus.

Latest Math Topics
Jun 26, 2023
Jun 23, 2023

Latest Math Problems
Jul 01, 2023
Jun 25, 2023
###### Math Questions

The math problems with solutions to learn how to solve a problem.

Learn solutions

Practice now

###### Math Videos

The math videos tutorials with visual graphics to learn every concept.

Watch now

###### Subscribe us

Get the latest math updates from the Math Doubts by subscribing us.