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 $C = -\ln a + C'$

$\blacksquare$