# Primitive of Reciprocal of x squared by x squared minus a squared

## Theorem

$\displaystyle \int \frac {\mathrm d x} {x^2 \left({x^2 - a^2}\right)} = \frac 1 {a^2 x} + \frac 1 {2 a^3} \ln \left({\frac {x - a} {x + a} }\right) + C$

for $x^2 > a^2$.

## Proof

 $\displaystyle \int \frac {\mathrm d x} {x^2 \left({x^2 - a^2}\right)}$ $=$ $\displaystyle \int \left({\frac 1 {a^2 \left({x^2 - a^2}\right)} - \frac 1 {a^2 x^2} }\right) \ \mathrm d x$ Partial Fraction Expansion $\displaystyle$ $=$ $\displaystyle \frac 1 {a^2} \int \frac {\mathrm d x} {x^2 - a^2} - \frac 1 {a^2} \int \frac {\mathrm d x} {x^2}$ Linear Combination of Integrals $\displaystyle$ $=$ $\displaystyle \frac 1 {a^2} \int \frac {\mathrm d x} {x^2 - a^2} + \frac 1 {a^2 x} + C$ Primitive of Power $\displaystyle$ $=$ $\displaystyle \frac 1 {a^2} \left({\frac 1 2 \ln \left ({\frac {x - a} {x + a} }\right) }\right) + \frac 1 {a^2 x} + C$ Primitive of $\dfrac 1 {x^2 - a^2}$ $\displaystyle$ $=$ $\displaystyle \frac 1 {a^2 x} + \frac 1 {2 a^3} \ln \left({\frac {x - a} {x + a} }\right) + C$ simplifying

$\blacksquare$