The Definite Integral of sin^4x from 0 to pi/4

In this tutorial we shall derive the definite integral of the trigonometric function $\sin^{4}x$ from limits 0 to Pi/4.

The integration of the form is

Using the half angle formula from trigonometry ${\sin ^2}x = \frac{{1 - \cos 2x}}{2}$, we have

Again, using the half angle formula from trigonometry ${\cos ^2}2x = \frac{{1 + \cos 4x}}{2}$, we have

First we evaluate this integration by using the integral formula $\int {\cos kxdx = \frac{{\sin kx}}{k}}$, 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