# Primitive of x over a squared minus x squared

## Theorem

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

for $x^2 < a^2$.

## Proof

Let:

 $\ds z$ $=$ $\ds a^2 - x^2$ $\ds \leadsto \ \$ $\ds \frac {\d z} {\d x}$ $=$ $\ds -2 x$ Power Rule for Derivatives $\ds \leadsto \ \$ $\ds \int \frac {\d x} {x \paren {a^2 - x^2} }$ $=$ $\ds \int \frac {\d z} {-2 z}$ Integration by Substitution $\ds$ $=$ $\ds -\frac 1 2 \int \frac {\d z} z$ Primitive of Constant Multiple of Function $\ds$ $=$ $\ds -\frac 1 2 \ln z + C$ Primitive of Reciprocal: Corollary as $z > 0$ $\ds$ $=$ $\ds -\frac 1 2 \, \map \ln {a^2 - x^2} + C$ substituting for $z$

$\blacksquare$