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

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

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

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

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

Summarising:

Hence the result.