Primitive of Reciprocal of Hyperbolic Cosine of a x
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Theorem
- $\ds \int \frac {\d x} {\cosh a x} = \frac {2 \map \arctan {e^{a x} } } a + C$
Proof
\(\ds \int \frac {\d x} {\cosh a x}\) | \(=\) | \(\ds \int \sech a x \rd x\) | Definition 2 of Hyperbolic Secant | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {2 \map \arctan {e^{a x} } } a + C\) | Primitive of $\sech a x$: Arctangent of Exponential Form |
$\blacksquare$
Also see
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\cosh a x$: $14.567$