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}