Primitive of Hyperbolic Sine of a x by Hyperbolic Cosine of a x/Proof 4
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Theorem
- $\ds \int \sinh a x \cosh a x \rd x = \frac {\sinh^2 a x} {2 a} + C$
Proof
With a view to expressing the primitive in the form:
- $\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
\(\ds u\) | \(=\) | \(\ds \sinh a x\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac {\d u} {\d x}\) | \(=\) | \(\ds a \cosh a x\) | Derivative of $\sinh a x$ |
and let:
\(\ds \frac {\d v} {\d x}\) | \(=\) | \(\ds \cosh a x\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds v\) | \(=\) | \(\ds \frac {\sinh a x} a\) | Primitive of $\cosh a x$ |
Then:
\(\ds \int \sinh a x \cosh a x \rd x\) | \(=\) | \(\ds \paren {\sinh a x} \paren {\frac {\sinh a x} a} - \int \paren {\frac {\sinh a x} a} \paren {a \cosh a x} \rd x + C\) | Integration by Parts | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\sinh^2 a x} a - \int \sinh a x \cosh a x \rd x + C\) | simplifying | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds 2 \int \sinh a x \cosh a x \rd x\) | \(=\) | \(\ds \frac {\sinh^2 a x} a + C\) | gathering terms | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \int \sinh a x \cosh a x \rd x\) | \(=\) | \(\ds \frac {\sinh^2 a x} {2 a} + C\) | simplifying |
$\blacksquare$