Primitive of Square of Hyperbolic Cosine Function/Corollary

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Corollary to Primitive of Square of Hyperbolic Cosine Function

$\displaystyle \int \cosh^2 x \ \mathrm d x = \frac {\sinh x \cosh x + x} 2 + C$

where $C$ is an arbitrary constant.


Proof

\(\displaystyle \int \cosh^2 x \ \mathrm d x\) \(=\) \(\displaystyle \frac {\sinh 2 x} 4 + \frac x 2 + C\) Primitive of Square of Hyperbolic Cosine Function
\(\displaystyle \) \(=\) \(\displaystyle \frac {2 \sinh x \cosh x} 4 + \frac x 2 + C\) Double Angle Formula for Hyperbolic Sine
\(\displaystyle \) \(=\) \(\displaystyle \frac {\sinh x \cosh x + x} 2 + C\) rearranging

$\blacksquare$


Sources