Primitive of Hyperbolic Secant of a x/Arctangent of Hyperbolic Sine Form

From ProofWiki
Jump to navigation Jump to search

Theorem

$\ds \int \sech a x \rd x = \frac {\map \arctan {\sinh a x} } a + C$


Proof

\(\ds \int \sech x \rd x\) \(=\) \(\ds \map \arctan {\sinh x}\) Primitive of $\sech x$: Arctangent of Hyperbolic Sine form
\(\ds \leadsto \ \ \) \(\ds \int \sech a x \rd x\) \(=\) \(\ds \frac 1 a \map \arctan {\sinh a x} + C\) Primitive of Function of Constant Multiple
\(\ds \) \(=\) \(\ds \frac {\map \arctan {\sinh a x} } a + C\) simplifying

$\blacksquare$