The Definite Integral Inverse Tangent from 0 to Pi over 4

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

The integration of the form is

Here the first function ${\tan ^{ - 1}}x$ and the second function will be $1$. Using the formula for integration by parts in definite integral form, we have

Equation (i) becomes:

Using the basic rule of 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