Primitive of Hyperbolic Secant of a x/Arctangent of Hyperbolic Sine Form
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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 |
$\blacksquare$