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

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Theorem

For $x \in \R_{\ne 0}$:

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


Proof

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

$\blacksquare$


Also presented as

This result is also seen presented in the form:

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


Also see


Sources

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