Primitive of Reciprocal of Root of x squared plus a squared/Logarithm Form

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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

\(\displaystyle \int \frac {\d x} {\sqrt {x^2 + a^2} }\) \(=\) \(\displaystyle \sinh^{-1} {\frac x a} + C\) Primitive of Reciprocal of $\sqrt {x^2 + a^2}$ in $\sinh^{-1}$ form
\(\displaystyle \) \(=\) \(\displaystyle \map \ln {x + \sqrt {x^2 + a^2} } - \ln a + C\) $\sinh^{-1} {\dfrac x a}$ in Logarithm Form
\(\displaystyle \) \(=\) \(\displaystyle \map \ln {x + \sqrt {x^2 + a^2} } + C\) subsuming $-\ln a$ into arbitrary constant

$\blacksquare$


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