# Primitive of x over x squared plus a squared squared

## Theorem

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

## Proof

Let:

 $\ds z$ $=$ $\ds x^2 + a^2$ $\ds \leadsto \ \$ $\ds \frac {\d z} {\d x}$ $=$ $\ds 2 x$ Derivative of Power $\ds \leadsto \ \$ $\ds \int \frac {x \rd x} {\paren {x^2 + a^2}^2}$ $=$ $\ds \int \frac {\d z} {2 z^2}$ Integration by Substitution $\ds$ $=$ $\ds -\frac 1 {2 z} + C$ Primitive of Power $\ds$ $=$ $\ds -\frac 1 {2 \paren {x^2 + a^2} } + C$ substituting for $z$

$\blacksquare$