Primitive of Hyperbolic Sine of a x by Hyperbolic Cosine of a x/Proof 1
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Theorem
- $\ds \int \sinh a x \cosh a x \rd x = \frac {\sinh^2 a x} {2 a} + C$
Proof
| \(\ds \int \sinh a x \cosh a x \rd x\) | \(=\) | \(\ds \int \frac {\sinh 2 a x} 2 \rd x\) | Double Angle Formula for Hyperbolic Sine | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 2 \int \sinh 2 a x \rd x\) | Primitive of Constant Multiple of Function | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 2 \paren {\frac {\cosh 2 a x} {2 a} } + C\) | Primitive of $\sinh a x$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 2 \paren {\frac {1 + 2 \sinh^2 a x} {2 a} } + C\) | Double Angle Formula for Hyperbolic Cosine: Corollary $2$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {\sinh^2 a x} {2 a} + \frac 1 {4 a} + C\) | simplifying | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {\sinh^2 a x} {2 a} + C\) | subsuming $\dfrac 1 {4 a}$ into arbitrary constant |
$\blacksquare$