# Primitive of Reciprocal of x squared minus a squared/Logarithm Form

## Theorem

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

where $x^2 > a^2$.

## Proof 1

 $\displaystyle \forall x \in \R: x^2 > a^2: \ \$ $\displaystyle \int \frac {\d x} {x^2 - a^2}$ $=$ $\displaystyle -\frac 1 a \coth^{-1} {\frac x a} + C$ Primitive of Reciprocal of $x^2 - a^2$ in $\coth^{-1}$ form $\displaystyle$ $=$ $\displaystyle -\frac 1 a \paren {\dfrac 1 2 \map \ln {\dfrac {x + a} {x - a} } } + C$ $\coth^{-1} {\dfrac x a}$ in Logarithm Form $\displaystyle$ $=$ $\displaystyle -\frac 1 {2 a} \map \ln {\frac {x + a} {x - a} } + C$ simplifying $\displaystyle$ $=$ $\displaystyle \frac 1 {2 a} \map \ln {\frac {x - a} {x + a} } + C$ Logarithm of Reciprocal

$\blacksquare$

## Proof 2

 $\displaystyle \int \frac {\d x} {x^2 - a^2}$ $=$ $\displaystyle \int \frac {\d x} {\paren {x - a} \paren {x + a} }$ Difference of Two Squares $\displaystyle$ $=$ $\displaystyle \int \frac {\d x} {2 a \paren {x - a} } - \int \frac {\d x} {2 a \paren {x + a} }$ Partial Fraction Expansion $\displaystyle$ $=$ $\displaystyle \frac 1 {2 a} \int \frac {\d x} {x - a} - \frac 1 {2 a} \int \frac {\d x} {x + a}$ Primitive of Constant Multiple of Function $\displaystyle$ $=$ $\displaystyle \frac 1 {2 a} \ln \size {x - a} - \frac 1 {2 a} \ln \size {x + a} + C$ Primitive of Reciprocal $\displaystyle$ $=$ $\displaystyle \dfrac 1 {2 a} \ln \size {\dfrac {x - a} {x + a} } + C$ Difference of Logarithms

$\blacksquare$