Primitive of Reciprocal of x by x squared minus a squared

Theorem

$\displaystyle \int \frac {\d x} {x \paren {x^2 - a^2} } = \frac 1 {2 a^2} \, \map \ln {\frac {x^2 - a^2} {x^2} } + C$

for $x^2 > a^2$.

Proof

 $\ds \int \frac {\d x} {x \paren {x^2 - a^2} }$ $=$ $\ds \int \paren {\frac x {a^2 \paren {x^2 - a^2} } - \frac 1 {a^2 x} } \rd x$ Partial Fraction Expansion $\ds$ $=$ $\ds \frac 1 {a^2} \int \frac {x \rd x} {x^2 - a^2} - \frac 1 {a^2} \int \frac {\d x} x$ Linear Combination of Integrals $\ds$ $=$ $\ds \frac 1 {a^2} \int \frac {x \rd x} {x^2 - a^2} - \frac 1 {a^2} \ln \size x + C$ Primitive of Reciprocal $\ds$ $=$ $\ds \frac 1 {a^2} \paren {\frac 1 2 \map \ln {x^2 - a^2} } - \frac 1 {a^2} \ln \size x + C$ Primitive of $\dfrac x {x^2 - a^2}$ $\ds$ $=$ $\ds \frac 1 {2 a^2} \map \ln {x^2 - a^2} - \frac 1 {2 a^2} \ln \size {x^2} + C$ Logarithm of Power $\ds$ $=$ $\ds \frac 1 {2 a^2} \map \ln {x^2 - a^2} - \frac 1 {2 a^2} \map \ln {x^2} + C$ as $x^2 > 0$ $\ds$ $=$ $\ds \frac 1 {2 a^2} \map \ln {\frac {x^2 - a^2} {x^2} } + C$ Difference of Logarithms

$\blacksquare$