Primitive of Hyperbolic Sine of a x by Hyperbolic Cosine of a x/Proof 5
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Theorem
- $\ds \int \sinh a x \cosh a x \rd x = \frac {\sinh^2 a x} {2 a} + C$
Proof
| \(\ds u\) | \(=\) | \(\ds \sinh a x\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \frac {\d u} {\d x}\) | \(=\) | \(\ds a \cosh a x\) | Derivative of $\sinh a x$ | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \int \sinh a x \cosh a x \rd x\) | \(=\) | \(\ds \int \frac u a \rd u\) | Integration by Substitution | ||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 a \int u \rd u\) | Primitive of Constant Multiple of Function | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 a \frac {u^2} 2 + C\) | Primitive of Power | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {\sinh^2 a x} {2 a} + C\) | substituting for $u$ |
$\blacksquare$