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
- 1968: George B. Thomas, Jr.: Calculus and Analytic Geometry (4th ed.) ... (previous) ... (next): Back endpapers: A Brief Table of Integrals: $129$.