# Primitive of Reciprocal of x by Root of x squared plus a squared/Logarithm Form

## Theorem

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

## Proof

 $\displaystyle \int \frac {\d x} {x \sqrt {x^2 + a^2} }$ $=$ $\displaystyle -\frac 1 a \csch^{-1} {\frac x a} + C$ Primitive of Reciprocal of $x \sqrt {x^2 + a^2}$: $\csch^{-1}$ form $\displaystyle$ $=$ $\displaystyle -\frac 1 a \map \ln {\frac {a + \sqrt{a^2 + x^2} } x} + C$ $\csch^{-1} {\dfrac x a}$ in Logarithm Form

$\blacksquare$