# Primitive of Reciprocal of Root of x squared minus a squared/Inverse Hyperbolic Cosine Form

## Theorem

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

## Proof

Let:

 $\displaystyle u$ $=$ $\displaystyle \cosh^{-1} {\frac x a}$ $\displaystyle x$ $=$ $\displaystyle a \cosh u$ $\displaystyle \leadsto \ \$ $\displaystyle \frac {\d x} {\d u}$ $=$ $\displaystyle a \sinh u$ Derivative of Hyperbolic Cosine Function $\displaystyle \leadsto \ \$ $\displaystyle \int \frac {\d x} {\sqrt {x^2 - a^2} }$ $=$ $\displaystyle \int \frac {a \sinh u} {\sqrt {a^2 \cosh^2 u - a^2} } \rd u$ Integration by Substitution $\displaystyle$ $=$ $\displaystyle \frac a a \int \frac {\sinh u} {\sqrt {\cosh^2 u - 1} } \rd u$ Primitive of Constant Multiple of Function $\displaystyle$ $=$ $\displaystyle \int \frac {\sinh u} {\sinh u} \rd u$ Difference of Squares of Hyperbolic Cosine and Sine $\displaystyle$ $=$ $\displaystyle \int 1 \rd u$ $\displaystyle$ $=$ $\displaystyle u + C$ Integral of Constant $\displaystyle$ $=$ $\displaystyle \cosh^{-1} {\frac x a} + C$ Definition of $u$

$\blacksquare$