Primitive of Hyperbolic Sine of a x by Hyperbolic Cosine of a x/Proof 2
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Theorem
- $\ds \int \sinh a x \cosh a x \rd x = \frac {\sinh^2 a x} {2 a} + C$
Proof
\(\ds \int \sinh a x \cosh a x \rd x\) | \(=\) | \(\ds \int \cosh a x \sinh a x \rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\cosh^2 a x} {2 a} + C\) | Primitive of $\cosh^n a x \sinh a x$ using $n = 1$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {1 + \sinh^2 a x} {2 a} + C\) | Difference of Squares of Hyperbolic Cosine and Sine | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {2 a} + \frac {\sinh^2 a x} {2 a} + C\) | simplifying | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\sinh^2 a x} {2 a} + C\) | subsuming $\dfrac 1 {2 a}$ into arbitrary constant |
$\blacksquare$