# Primitive of Square of Hyperbolic Cosine Function

## 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

$\blacksquare$