# Primitive of Reciprocal of x by Root of x squared plus a squared/Inverse Hyperbolic Cosecant Form

## Theorem

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

## Proof

Let:

 $\ds u$ $=$ $\ds \csch^{-1} {\frac x a}$ $\ds \leadsto \ \$ $\ds x$ $=$ $\ds a \csch u$ $\ds \leadsto \ \$ $\ds \frac {\d x} {\d u}$ $=$ $\ds -a \csch u \coth u$ Derivative of Hyperbolic Cosecant $\ds \leadsto \ \$ $\ds \int \frac {\d x} {x \sqrt {x^2 + a^2} }$ $=$ $\ds \int \frac {-a \csch u \coth u} {a \csch u \sqrt {a^2 \csch^2 u + a^2} } \rd u$ Integration by Substitution $\ds$ $=$ $\ds -\frac a {a^2} \int \frac {\csch u \coth u} {\csch u \sqrt {\csch^2 u - 1} } \rd u$ Primitive of Constant Multiple of Function $\ds$ $=$ $\ds -\frac 1 a \int \frac {\csch u \coth u} {\csch u \coth u} \rd u$ Difference of Squares of Hyperbolic Cotangent and Cosecant $\ds$ $=$ $\ds -\frac 1 a \int 1 \rd u$ $\ds$ $=$ $\ds -\frac 1 a u + C$ Integral of Constant $\ds$ $=$ $\ds \frac 1 a \csch^{-1} {\frac x a} + C$ Definition of $u$

$\blacksquare$