Primitive of Power of Hyperbolic Sine of a x by Hyperbolic Cosine of a x
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Theorem
- $\ds \int \sinh^n a x \cosh a x \rd x = \frac {\sinh^{n + 1} a x} {\paren {n + 1} a} + C$
for $n \ne -1$.
Proof
\(\ds u\) | \(=\) | \(\ds \sinh a x\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac {\d u} {\d x}\) | \(=\) | \(\ds a \cosh a x\) | Derivative of $\sinh a x$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \int \sinh^n a x \cosh a x \rd x\) | \(=\) | \(\ds \int \frac {u^n} a \rd u\) | Integration by Substitution | ||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 a \int u^n \rd u\) | Primitive of Constant Multiple of Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 a \frac {u^{n + 1} } {n + 1} + C\) | Primitive of Power | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\sinh^{n + 1} a x} {\paren {n + 1} a} + C\) | substituting for $u$ |
$\blacksquare$
Also see
- Primitive of $\coth a x$ for the case where $n = -1$.
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\sinh a x $ and $\cosh a x$: $14.592$