Primitive of Hyperbolic Sine of p x by Hyperbolic Cosine of q x
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Theorem
- $\ds \int \sinh p x \cosh q x \rd x = \frac {\map \cosh {p + q} x} {2 \paren {p + q} } + \frac {\map \cosh {p - q} x} {2 \paren {p - q} } + C$
Proof
| \(\ds \int \sinh p x \cosh q x \rd x\) | \(=\) | \(\ds \int \paren {\frac {\map \sinh {p x + q x} + \map \sinh {p x - q x} } 2} \rd x\) | Werner Formula for Hyperbolic Sine by Hyperbolic Cosine | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 2 \int \map \sinh {p + q} x \rd x + \frac 1 2 \int \map \sinh {p - q} x \rd x\) | Linear Combination of Primitives | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 2 \frac {\map \cosh {p + q} x} {p + q} + \frac 1 2 \frac {\map \cosh {p - q} x} {p - q} + C\) | Primitive of $\sinh a x$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {\map \cosh {p + q} x} {2 \paren {p + q} } + \frac {\map \cosh {p - q} x} {2 \paren {p - q} } + C\) | simplifying |
$\blacksquare$
Also see
- Primitive of $\sinh a x \sinh p x$
- Primitive of $\sinh a x \cosh a x$
- Primitive of $\cosh a x \cosh p x$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\sinh a x $ and $\cosh a x$: $14.591$
- 2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 17$: Tables of Special Indefinite Integrals: $(29)$ Integrals Involving $\sinh a x $ and $\cosh a x$: $17.29.2.$