Integral of Hyperbolic Secant Squared

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

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

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

Other Integral Formulae of the Hyperbolic Secant Square Function

The other formulae of the hyperbolic secant square integral with an angle of hyperbolic sine in the form of a function are:


\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}