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

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Theorem

$\displaystyle \int \frac {\d x} {x \sqrt {x^2 + a^2} } = -\frac 1 a \map \ln {\frac {a + \sqrt {x^2 + a^2} } x} + C$


Proof

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

$\blacksquare$



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