# Integration of Secx Tanx

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

The integration of secant tangent is of the form

To prove this formula, consider

Using the derivative formula $\frac{d}{{dx}}\sec x = \sec x\tan x$, 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 tangent integral with an  angle in the form of a function are given as

1.

2.

Example: Evaluate the integral $\int {\sec 5x\tan 5xdx}$ with respect to $x$

We have integral

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