# Primitive of x over x squared plus a squared

## Theorem

$\ds \int \frac {x \rd x} {x^2 + a^2} = \frac 1 2 \map \ln {x^2 + a^2} + C$

where $a$ is a non-zero constant.

## Proof 1

 $\ds u$ $=$ $\ds x^2 + a^2$ $\ds \leadsto \ \$ $\ds \frac {\d u} {\d x}$ $=$ $\ds 2 x$ Power Rule for Derivatives and Derivative of Constant $\ds \leadsto \ \$ $\ds \int \frac {x \rd x} {x^2 + a^2}$ $=$ $\ds \frac 1 2 \ln \size {x^2 + a^2} + C$ Primitive of Function under its Derivative $\ds$ $=$ $\ds \frac 1 2 \, \map \ln {x^2 + a^2} + C$ Absolute Value of Even Power‎

$\blacksquare$

## Proof 2

$\ds \int \frac {x^{n - 1} \rd x} {x^n + a^n} = \frac 1 n \ln \size {x^n + a^n} + C$

So:

 $\ds \int \frac {x \rd x} {x^2 + a^2}$ $=$ $\ds \frac 1 2 \ln \size {x^2 + a^2} + C$ Primitive of $\dfrac {x^{n - 1} } {\paren {x^n + a^n} }$ with $n = 2$ $\ds$ $=$ $\ds \frac 1 2 \, \map \ln {x^2 + a^2} + C$ Absolute Value of Even Power

$\blacksquare$