Primitive of Square of Hyperbolic Cosine Function

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Theorem

$\displaystyle \int \cosh^2 x \ \mathrm d x = \frac {\sinh 2 x} 4 + \frac x 2 + C$

where $C$ is an arbitrary constant.


Corollary

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

where $C$ is an arbitrary constant.


Proof

\(\displaystyle \int \cosh^2 x \ \mathrm d x\) \(=\) \(\displaystyle \int \left({\frac {\cosh 2 x + 1} 2}\right) \ \mathrm d x\) Square of Hyperbolic Cosine
\(\displaystyle \) \(=\) \(\displaystyle \int \left({\frac {\cosh 2 x} 2}\right) \ \mathrm d x + \int \frac 1 2 \ \mathrm d x\) Linear Combination of Integrals
\(\displaystyle \) \(=\) \(\displaystyle \int \left({\frac {\cosh 2 x} 2}\right) \ \mathrm d x + \frac x 2 + C\) Primitive of Constant
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 2 \left({\frac {\sinh \left({2 x}\right)} 2}\right) + \frac x 2 + C\) Primitive of Function of Constant Multiple and Primitive of Hyperbolic Cosine Function
\(\displaystyle \) \(=\) \(\displaystyle \frac {\sinh 2 x} 4 + \frac x 2 + C\) rearranging

$\blacksquare$


Sources