# Primitive of Reciprocal of Root of x squared minus a squared/Logarithm Form/Proof 1

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## Theorem

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

## Proof

\(\displaystyle \int \frac {\d x} {\sqrt {x^2 - a^2} }\) | \(=\) | \(\displaystyle \cosh^{-1} {\frac x a} + C'\) | Primitive of Reciprocal of $\sqrt {x^2 - a^2}$: $\cosh^{-1}$ form | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \map \ln {\frac x a + \sqrt {\paren {\frac x a}^2 - 1} } + C'\) | Definition of Real Inverse Hyperbolic Cosine | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \map \ln {\frac x a + \sqrt {\frac {x^2 - a^2} {a^2} } } + C'\) | rearrangement | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \map \ln {\frac {x + \sqrt {x^2 - a^2} } a} + C'\) | rearrangement | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \map \ln {x + \sqrt {x^2 - a^2} } - \ln a + C'\) | Difference of Logarithms | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \map \ln {x + \sqrt {x^2 - a^2} } + C\) | putting $C = -\ln a + C'$ |

$\blacksquare$