Primitive of Reciprocal of a squared minus x squared

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Theorem

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

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


Inverse Hyperbolic Function Form

$\ds \int \dfrac {\d x} {a^2 - x^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} {a^2 - x^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

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


Also see