`int_(0)^(pi//2) sin^(2)x.cos^(3)x.dx` का मान ज्ञात करे।

Explanation

`int_(0)^(pi//2) sin^(2)x . Cos^(3)x.dx`
`=int_(0)^(pi//2)x.cos^(2)x.cosx.dx`
`int_(0)^(pi//2) sin^(2)x(1 - sin^(2)x) cos , dx`
माना की sin x = z
`therefore cos x. dx = dz`
जब x = 0, तब z = 0
और जब `x = (pi)/(2), z = (pi)/(2) = 1`
`int_(0)^(pi//2) sin^(2) cos^(3) x dx = int_(0)^(1) z^(2) (1 - z^(2)) dx`
`=[(z^(3))/(3)-(z^(5))/(5)]_(0)^(1)`
`=[(1^(3)=-0^(3))/(3)-(1^(5)-0^(5))/(5)]`
`=(1)/(3)-(1)/(5)=(5-3)/(15)=(2)/(15)`