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 $u$

$\blacksquare$


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