Integral of Hyperbolic Sine

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

The integration of the hyperbolic sine function is of the form

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

To prove this formula, consider

\frac{d}{{dx}}\left[ {\cosh x + c} \right] = \frac{d}{{dx}}\cosh x + \frac{d}{{dx}}c

Using the derivative formula \frac{d}{{dx}}\cosh x = \sinh x, we have

\begin{gathered} \frac{d}{{dx}}\left[ {\cosh x + c} \right] = \frac{d}{{dx}}\cosh x + \frac{d}{{dx}}c \\ \Rightarrow \frac{d}{{dx}}\left[ {\cosh x + c} \right] = \sinh x + 0 \\ \Rightarrow \frac{d}{{dx}}\left[ {\cosh x + c} \right] = \sinh x \\ \Rightarrow \sinh x = \frac{d}{{dx}}\left[ {\cosh x + c} \right] \\ \Rightarrow \sinh xdx = d\left[ {\cosh x + c} \right]\,\,\,\,{\text{ - - - }}\left( {\text{i}} \right) \\ \end{gathered}

Integrating both sides of equation (i) with respect to x, we have

\int {\sinh xdx} = \int {d\left[ {\cosh 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 {\sinh xdx = } \cosh x + c

Other Integral Formulae of the Hyperbolic Sine Function

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

1.

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

2.

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