Show that `int_0^(pi/2)sqrt((sin2theta))sinthetadtheta=pi/4`

Explanation

Let `I=int_(0)^(pi//2)sqrt((sin 2theta))sin theta d theta `……………1
`:.I=int_(0)^(pi//2)sqrt(sin(2(pi/2)-theta))sin(pi/2-theta)d theta`
`=int_(0)^(pi//2)sqrt((sin 2theta))cos theta d theta` …………..2
Adding 1 and 2 we get
`2I=int_(0)^(pi//2)sqrt((sin 2theta))(sin theta+costheta)d theta`
or `I=1/2int_(0)^(pi//2)sqrt(1-(sin theta -cos theta)^(2))(sin theta+cos theta) d theta`
`=1/2int_(-1)^(1)sqrt(1-t^(2))dt`[ Let `sin theta -cos theta=t`]
`=1/2[1/2 t sqrt((1-t^(2)))+1/2sin^(-1)t]_(-1)^(1)=(pi)/4`