Integral of Hyperbolic Cosecant Squared

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

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

\int {{{\operatorname{csch} }^2}xdx = } - \coth x + c

Other Integral Formulas of Hyperbolic Cosecant Square Function:

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

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}