Question

Solve : $\underset{0}{\stackrel{\frac{\pi }{2}}{\int }}si{n}^{2}xdx$

Answer 1

Let $I={\int }_0^{\frac{\pi }{2}}si{n}^{2}xdx$ .................. (1)

By applying ${\int }_0^{a}f\left(x\right)dx={\int }_0^{a}f(a-x)dx$

$I={\int }_0^{\frac{\pi }{2}}si{n}^2(\frac{\pi }{2}-x)dx$

$I={\int }_0^{\frac{\pi }{2}}co{s}^2\left(x\right)dx$ .............. (2)

Add (1) and (2) ,

$2I={\int }_0^{\frac{\pi }{2}}(si{n}^{2}x+co{s}^{2}x)dx$

$2I={\int }_0^{\frac{\pi }{2}}1dx$

$2I={\left[x\right]}_0^{\frac{\pi }{2}}$

$2I=\frac{\pi }{2}$

$I=\frac{\pi }{4}$