# Integral of Hyperbolic Cosine

The integration of the hyperbolic cosine function is an important integral formula in integral calculus. This integral belongs to the hyperbolic formulae.

The integration of the hyperbolic cosine function is of the form
$\int {\cosh xdx = } \sinh x + c$

To prove this formula, consider
$\frac{d}{{dx}}\left[ {\sinh x + c} \right] = \frac{d}{{dx}}\sinh x + \frac{d}{{dx}}c$

Using the derivative formula $\frac{d}{{dx}}\sinh x = \cosh x$, we have
$\begin{gathered} \frac{d}{{dx}}\left[ {\sinh x + c} \right] = \frac{d}{{dx}}\sinh x + \frac{d}{{dx}}c \\ \Rightarrow \frac{d}{{dx}}\left[ {\sinh x + c} \right] = \cosh x + 0 \\ \Rightarrow \frac{d}{{dx}}\left[ {\sinh x + c} \right] = \cosh x \\ \Rightarrow \cosh x = \frac{d}{{dx}}\left[ {\sinh x + c} \right] \\ \Rightarrow \cosh xdx = d\left[ {\sinh x + c} \right]\,\,\,\,{\text{ – – – }}\left( {\text{i}} \right) \\ \end{gathered}$

Integrating both sides of equation (i) with respect to $x$, we have
$\int {\cosh xdx} = \int {d\left[ {\sinh x + c} \right]}$

As we know that by definition integration is the inverse process of the derivative, so the integral sign $\int {}$ and $\frac{d}{{dx}}$ on the right side will cancel each other out, i.e.
$\int {\cosh xdx = } \sinh x + c$

Other Integral Formulae of the Hyperbolic Cosine Function

The other formulae of the hyperbolic cosine integral with the angle of hyperbolic cosine in the form of a function are:

1. $\int {\cosh axdx = \frac{{\sinh ax}}{a}} + c$

2. $\int {\cosh f\left( x \right)f’\left( x \right)dx = \sinh f\left( x \right) + c}$