Primitive of Hyperbolic Cosine of a x by Hyperbolic Cosine of p x
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Theorem
- $\ds \int \cosh a x \cosh p x \rd x = \frac {\map \sinh {a + p} x} {2 \paren {a + p} } + \frac {\map \sinh {a - p} x} {2 \paren {a - p} } + C$
Proof
| \(\ds \int \cosh a x \cosh p x \rd x\) | \(=\) | \(\ds \int \paren {\frac {\map \cosh {a x + p x} + \map \cosh {a x - p x} } 2} \rd x\) | Werner Formula for Hyperbolic Cosine by Hyperbolic Cosine | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 2 \int \map \cosh {a + p} x \rd x + \frac 1 2 \int \map \cosh {a - p} x \rd x\) | Linear Combination of Primitives | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 2 \frac {\map \sinh {a + p} x} {a + p} + \frac 1 2 \frac {\map \sinh {a - p} x} {a - p} + C\) | Primitive of $\cosh a x$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {\map \sinh {a + p} x} {2 \paren {a + p} } + \frac {\map \sinh {a - p} x} {2 \paren {a - p} } + C\) | simplifying |
$\blacksquare$
Also see
- Primitive of $\sinh a x \sinh p x$
- Primitive of $\sinh a x \cosh a x$
- Primitive of $\sinh p x \cosh q x$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\cosh a x$: $14.572$
- 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: $(28)$ Integrals Involving $\cosh a x$: $17.28.11.$