$\int \dfrac{dx}{a^2 - x^2} $

The correct answer is $\dfrac{1}{2a} log \left| \dfrac{a + x}{a - x} \right|$ + $C$

$\int \dfrac{dx}{a^2 - x^2} $

$\int \dfrac{dx}{(a - x)(a + x)} $

$\dfrac{1}{2a} \int \left[ \dfrac{(a - x) + (a + x)}{(a - x)(a + x)} \right] . dx$

$\dfrac{1}{2a} \left[ \int \dfrac{1}{(a + x)} dx + \int \dfrac{1}{(a - x)} dx \right] $

$\dfrac{1}{2a} \left[ log |a + x| - log|a - x| \right]$ + $C$

$\dfrac{1}{2a} log \left| \dfrac{a + x}{a - x} \right|$ + $C$