Primitive of Reciprocal of x squared minus a squared/Logarithm Form 2/Proof 2/Partial Fraction Expansion

Lemma for Primitive of Reciprocal of $\paren {x^2 - a^2}$

 * $\dfrac 1 {x^2 - a^2} \equiv \dfrac 1 {2 a \paren {x - a} } - \dfrac 1 {2 a \paren {x + a} }$

Proof
Setting $x = a$ in $(1)$:

Setting $x = -a$ in $(1)$:

Summarising:

Hence the result.