Integration of Sin Squared x

In this tutorial we shall drive the integral of sine squared x.

The integration of the form

I =  \int {{{\sin }^2}xdx}

This integral cannot be evaluated by direct formula of integration, so using trigonometric identity of half angle {\sin ^2}x = \frac{{1 - \cos 2x}}{2}, we have

\begin{gathered} I = \int {\left( {\frac{{1 - \cos 2x}}{2}}  \right)dx} \\ \Rightarrow I = \frac{1}{2}\int {\left( {1 -  \cos 2x} \right)dx} \\ \Rightarrow I = \frac{1}{2}\int {1dx -  \frac{1}{2}\int {\cos 2xdx} } \\ \end{gathered}

Using the integral formula \int {\cos kxdx = \frac{{\sin kx}}{k} + c} , we have

\begin{gathered} \int {{{\sin }^2}x} dx = \frac{1}{2}x -  \frac{1}{2}\frac{{\sin 2x}}{2} + c \\ \Rightarrow \int {{{\sin }^2}x} dx =  \frac{1}{2}x - \frac{1}{4}\sin 2x + c \\ \end{gathered}