निम्न फलनों के मान ज्ञात कीजिए -
`int_(0)^(pi)x sin^(2)xdx`

Explanation

माना `I=int_(0)^(pi)x sin^(2)x dx`
`=(1)/(2)int_(0)^(pi)x(1-cos2x)dx`
`=(1)/(2)int_(0)^(pi)x dx-(1)/(2)int_(0)^(pi)x cos 2x dx`
`=[(1)/(4)x^(2)]_(0)^(pi)-(1)/(2)[{x(sin 2x)/(2)}_(0)^(pi)-int_(0)^(pi)1(sin2x)/(2)dx]`
`=(1)/(4)[pi^(2)-0]-(1)/(2)[{x(sin2x)/(2)}_(0)^(pi)+{(cos 2x)/(4)}_(0)^(pi)]`
`=(1)/(4)[pi^(4)-0]-(1)/(2)[(pi sin 2pi)/(2)-(0sin0)/(2)]-(1)/(8)[cos 2pi-cos 0]`
`=(pi^(2))/(4)`