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}

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