Evaluating the integral ${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}si{n}^{2}xdx$ requires two steps.

First, write

$si{n}^{2}x=(sinx)(sinx)$

and integrate by parts to show that

${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}si{n}^{2}xdx=-sinxcosx+{\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}co{s}^{2}xdx$

Then, use $si{n}^{2}x+co{s}^{2}x=1$ to replace $co{s}^{2}x$ in the integral on the right$-$hand side, and complete the integration of ${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}si{n}^{2}xdx.$