# Integration of Secant Squared X

Integration of secant squared of x is an important integral formula in integral calculus, and this integral belongs to the trigonometric formulae.

The integration of secant squared of x is of the form

To prove this formula, consider

Using the derivative formula $\frac{d}{{dx}}\tan x = {\sec ^2}x$, 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, the integral sign $\int {}$and $\frac{d}{{dx}}$ on the right side will cancel each other out, i.e.

Other Integral Formulae of the Secant Tangent Function

The other formulae of secant squared of x integral with an angle in the form of s function are given as

1.

2.

Example: Evaluate the integral $\int {{{\sec }^2}3xdx}$ with respect to $x$

We have integral

Using the formula $\int {{{\sec }^2}axdx = \frac{{\tan ax}}{a}} + c$, we have