Primitive of Hyperbolic Cosine of a x by Hyperbolic Cosine of p x

From ProofWiki
Jump to navigation Jump to search

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


Sources