Primitive of Reciprocal of x squared minus a squared

Theorem

$\color {blue} {\dfrac 1 {x^2 - a^2} } \qquad \color {green} {-\dfrac 1 a \tanh^{-1} \dfrac x a} \qquad \color {red} {-\dfrac 1 a \coth^{-1} \dfrac x a}$

Let $a \in \R_{>0}$ be a strictly positive real constant.

Inverse Hyperbolic Function Form

$\ds \int \dfrac {\d x} {x^2 - a^2} = \begin {cases} -\dfrac 1 a \tanh^{-1} \dfrac x a + C & : \size x < a \\

& \\ -\dfrac 1 a \coth^{-1} \dfrac x a + C & : \size x > a \\ & \\ \text {undefined} & : x = a \end {cases}$

$1$st Logarithm Form

$\ds \int \frac {\d x} {x^2 - a^2} = \begin {cases} \dfrac 1 {2 a} \map \ln {\dfrac {a - x} {a + x} } + C & : \size x < a\\

& \\ \dfrac 1 {2 a} \map \ln {\dfrac {x - a} {x + a} } + C & : \size x > a \\ & \\ \text {undefined} & : \size x = a \end {cases}$

$2$nd Logarithm Form

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

Also see

• Primitive of $\dfrac 1 {x^2 + a^2}$
• Primitive of $\dfrac 1 {a^2 - x^2}$