Integral of Hyperbolic Cosine

Integration of hyperbolic cosine function is an important integral formula in integral calculus; this integral belongs to hyperbolic formulae.

The integration of 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 derivative, so the integral sign \int  {} and \frac{d}{{dx}} on the right side will cancel each other, i.e.

\int  {\cosh xdx = } \sinh x + c

Other Integral Formulas of Hyperbolic Cosine Function:

The other formulas of hyperbolic cosine integral with angle of hyperbolic cosine is in the form of function are given as

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


\int  {\cosh f\left( x \right)f'\left( x \right)dx = \sinh f\left( x \right) + c}