Primitive of Hyperbolic Secant of a x/Arcsine Form

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Theorem

$\ds \int \sech a x \rd x = \frac {\map \arcsin {\tanh a x} } a + C$


Proof

\(\ds \int \sech x \rd x\) \(=\) \(\ds \map \arcsin {\tanh x}\) Primitive of $\sech x$: Arcsine form
\(\ds \leadsto \ \ \) \(\ds \int \sech a x \rd x\) \(=\) \(\ds \frac 1 a \map \arcsin {\tanh a x} + C\) Primitive of Function of Constant Multiple
\(\ds \) \(=\) \(\ds \frac {\map \arcsin {\tanh a x} } a + C\) simplifying

$\blacksquare$


Sources