# 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

To prove this formula, consider

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

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

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.

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:

1.

2.