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

Theorem

Inverse Hyperbolic Cosecant Form

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

Inverse Hyperbolic Sine Form

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

Logarithm Form

For $x \in \R_{\ne 0}$:

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

Reciprocal Logarithm Form

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

Also see

• Primitive of Reciprocal of $x \sqrt {x^2 - a^2}$
• Primitive of Reciprocal of $x \sqrt {a^2 - x^2}$