Primitive of Reciprocal of x by Root of x squared plus a squared/Inverse Hyperbolic Cosecant Form

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



\(\ds u\) \(=\) \(\ds \csch^{-1} {\frac x a}\)
\(\ds \leadsto \ \ \) \(\ds x\) \(=\) \(\ds a \csch u\)
\(\ds \leadsto \ \ \) \(\ds \frac {\d x} {\d u}\) \(=\) \(\ds -a \csch u \coth u\) Derivative of Hyperbolic Cosecant
\(\ds \leadsto \ \ \) \(\ds \int \frac {\d x} {x \sqrt {x^2 + a^2} }\) \(=\) \(\ds \int \frac {-a \csch u \coth u} {a \csch u \sqrt {a^2 \csch^2 u + a^2} } \rd u\) Integration by Substitution
\(\ds \) \(=\) \(\ds -\frac a {a^2} \int \frac {\csch u \coth u} {\csch u \sqrt {\csch^2 u - 1} } \rd u\) Primitive of Constant Multiple of Function
\(\ds \) \(=\) \(\ds -\frac 1 a \int \frac {\csch u \coth u} {\csch u \coth u} \rd u\) Difference of Squares of Hyperbolic Cotangent and Cosecant
\(\ds \) \(=\) \(\ds -\frac 1 a \int 1 \rd u\)
\(\ds \) \(=\) \(\ds -\frac 1 a u + C\) Integral of Constant
\(\ds \) \(=\) \(\ds \frac 1 a \csch^{-1} {\frac x a} + C\) Definition of $u$


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