Integral of the Hyperbolic Cosecant Squared

In this tutorial we shall discuss the integration of the hyperbolic cosecant square function, and this integral is an important integral formula. This integral belongs to the hyperbolic formulae.

The integration of the hyperbolic cosecant square function is of the form
$\int {{{\operatorname{csch} }^2}xdx = – } \coth x + c$

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

Using the derivative formula $\frac{d}{{dx}}\coth x = – {\operatorname{csch} ^2}x$, we have
$\begin{gathered} \frac{d}{{dx}}\left[ { – \coth x + c} \right] = – \frac{d}{{dx}}\coth x + \frac{d}{{dx}}c \\ \Rightarrow \frac{d}{{dx}}\left[ { – \coth x + c} \right] = – \left( { – \csc {{\text{h}}^2}x} \right) + 0 \\ \Rightarrow \frac{d}{{dx}}\left[ { – \coth x + c} \right] = – \left( { – \csc {{\text{h}}^2}x} \right) \\ \Rightarrow \csc {{\text{h}}^2}x = \frac{d}{{dx}}\left[ { – \coth x + c} \right] \\ \Rightarrow \csc {{\text{h}}^2}xdx = d\left[ { – \coth x + c} \right]\,\,\,\,{\text{ – – – }}\left( {\text{i}} \right) \\ \end{gathered}$

Integrating both sides of equation (i) with respect to $x$, we have
$\int {{{\operatorname{csch} }^2}xdx} = \int {d\left[ { – \coth x + c} \right]}$

As we know that by definition integration is the inverse process of the derivative, so the integral signs $\int {}$and $\frac{d}{{dx}}$ on the right side will cancel each other out, i.e.
$\int {{{\operatorname{csch} }^2}xdx = } – \coth x + c$

Other Integral Formulae of the Hyperbolic Cosecant Square Function

The other formulae of the hyperbolic cosecant square integral with the angle of hyperbolic sine are:

1. $\int {{{\operatorname{csch} }^2}axdx = – \frac{{\coth ax}}{a}} + c$

2. $\int {{{\operatorname{csch} }^2}f\left( x \right)f’\left( x \right)dx = – \coth f\left( x \right) + c}$