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

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



\(\ds u\) \(=\) \(\ds \arsinh {\frac x a}\)
\(\ds \leadsto \ \ \) \(\ds x\) \(=\) \(\ds a \sinh u\) Definition of Real Area Hyperbolic Sine
\(\ds \leadsto \ \ \) \(\ds \frac {\d x} {\d u}\) \(=\) \(\ds a \cosh u\) Derivative of Hyperbolic Sine
\(\ds \leadsto \ \ \) \(\ds \int \frac {\d x} {\sqrt {x^2 + a^2} }\) \(=\) \(\ds \int \frac {a \cosh u} {\sqrt {a^2 \sinh^2 u + a^2} } \rd u\) Integration by Substitution
\(\ds \) \(=\) \(\ds \frac a a \int \frac {\cosh u} {\sqrt {\sinh^2 u + 1} } \rd u\) Primitive of Constant Multiple of Function
\(\ds \) \(=\) \(\ds \int \frac {\cosh u} {\cosh u} \rd u\) Difference of Squares of Hyperbolic Cosine and Sine
\(\ds \) \(=\) \(\ds \int 1 \rd u\)
\(\ds \) \(=\) \(\ds u + C\) Integral of Constant
\(\ds \) \(=\) \(\ds \arsinh {\frac x a} + C\) Definition of $u$


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