`int_(0)^(pi//4)sqrt(1+sin 2x)dx` का मान ज्ञात कीजिए ।

Explanation

`int_(0)^(pi//4)sqrt(1+sin2x)dx`
`=int_(0)^(pi//4)sqrt(sin^(@)x+cos^(2)x+2sin x cosx)`
`=int_(0)^(pi//4)sqrt((cosx+sinx)^(2))dx`
`(because 1=cos^(2)x+sin^(2)x, sin 2x=2sin x cosx)`
`=int_(0)^(pi//4)(cosx+sinx)dx=[sinx-cosx]_(0)^(pi//4)`
`=(sin.(pi)/(4)-cos.(pi)/(4))-(sin0-cos0)`
`=((1)/(sqrt2)-(1)/(sqrt2))-(0-1)=1`