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

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

The integration of the form is

First we evaluate this integration by using the integral formula $\int {\tan xdx = - \ln \cos x}$, and then we use the basic rule of the definite integral $\int\limits_a^b {f\left( x \right)dx = \left| {F\left( x \right)} \right|_a^b} = \left[ {F\left( b \right) - F\left( a \right)} \right]$. So we have

From the trigonometric values

and $\cos \frac{\pi }{4} = \frac{1}{{\sqrt 2 }}$, we have

Using the value $\ln 1 = 0$, we have