Primitive of Reciprocal of x by Root of x squared plus a squared

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Theorem

Inverse Hyperbolic Cosecant Form

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


Logarithm Form

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


Also see