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

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## 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 $u$ |

$\blacksquare$