Integral of Hyperbolic Secant Squared

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

The integration of hyperbolic secant square function is of the form

\int  {{{\operatorname{sech} }^2}xdx = } \tanh x + c

To prove this formula, consider

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

Using the derivative formula \frac{d}{{dx}}\tanh x = {\operatorname{sech} ^2}x, we have

\begin{gathered} \frac{d}{{dx}}\left[ {\tanh x + c} \right] =  \frac{d}{{dx}}\tanh x + \frac{d}{{dx}}c \\ \Rightarrow \frac{d}{{dx}}\left[ {\tanh x +  c} \right] = \sec {{\text{h}}^2}x + 0 \\ \Rightarrow \frac{d}{{dx}}\left[ {\tanh x +  c} \right] = \sec {{\text{h}}^2}x \\ \Rightarrow \sec {{\text{h}}^2}x =  \frac{d}{{dx}}\left[ {\tanh x + c} \right] \\ \Rightarrow \sec {{\text{h}}^2}xdx = d\left[  {\tanh x + c} \right]\,\,\,\,{\text{ -   -  - }}\left( {\text{i}} \right) \\ \end{gathered}

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

\int  {{{\operatorname{sech} }^2}xdx}  = \int {d\left[  {\tanh 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{sech} }^2}xdx = } \tanh x + c

Other Integral Formulas of Hyperbolic Secant Square Function:

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

\int  {{{\operatorname{sech} }^2}axdx = \frac{{\tanh ax}}{a}}  + c


\int  {{{\operatorname{sech} }^2}f\left( x \right)f'\left( x \right)dx = \tanh  f\left( x \right) + c}