# The Definite Integral of Secx Tanx from 0 to Pi over 4

In this tutorial we shall derive the definite integral of the trigonometric function secant tangent from limits 0 to Pi over 4.

The integration of the form is
$I = \int\limits_0^{\frac{\pi }{4}} {\sec x\tan xdx}$

First we evaluate this integration by using the integral formula $\int {\sec x\tan xdx = \sec x}$, and then we use the basic rule of the definite integral $\int\limits_a^b {f\left( x \right)dx = \left| {F\left( x \right)} \right|_a^b} = \left[ {F\left( b \right) – F\left( a \right)} \right]$. We have
$\begin{gathered} \int\limits_0^{\frac{\pi }{4}} {\sec x\tan xdx} = \left| {\sec x} \right|_0^{\frac{\pi }{4}} \\ \Rightarrow \int\limits_0^{\frac{\pi }{4}} {\sec x\tan xdx} = \left[ {\sec \frac{\pi }{4} – \sec 0} \right] \\ \end{gathered}$

From the trigonometric values $\sec \frac{\pi }{4} = \sqrt 2$ and $\sec 0 = 1$, we have
$\begin{gathered} \Rightarrow \int\limits_0^{\frac{\pi }{4}} {\sec x\tan xdx} = \left[ {\sqrt 2 – 1} \right] \\ \Rightarrow \int\limits_0^{\frac{\pi }{4}} {\sec x\tan xdx} = \sqrt 2 – 1 \\ \end{gathered}$