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

Lemma for Primitive of Reciprocal of $\left({a^2 - x^2}\right)$

 * $\dfrac 1 {a^2 - x^2} \equiv \dfrac 1 {2 a \left({a + x}\right)} + \dfrac 1 {2 a \left({a - x}\right)}$

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

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

Summarising:

Hence the result.