Integral of Hyperbolic Sine

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

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

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

Other Integral Formulas of Hyperbolic Sine Function:
The other formulas of hyperbolic sine integral with angle of hyperbolic sine is in the form of function are given as

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


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