Primitive of Square of Hyperbolic Cosine Function

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Theorem

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

where $C$ is an arbitrary constant.


Corollary

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

where $C$ is an arbitrary constant.


Proof

\(\displaystyle \int \cosh^2 x \rd x\) \(=\) \(\displaystyle \int \paren {\frac {\cosh 2 x + 1} 2} \rd x\) Square of Hyperbolic Cosine
\(\displaystyle \) \(=\) \(\displaystyle \int \paren {\frac {\cosh 2 x} 2} \rd x + \int \frac 1 2 \rd x\) Linear Combination of Integrals
\(\displaystyle \) \(=\) \(\displaystyle \int \paren {\frac {\cosh 2 x} 2} \rd x + \frac x 2 + C\) Primitive of Constant
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 2 \paren {\frac {\map \sinh {2 x} } 2} + \frac x 2 + C\) 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