Evaluate `int xsin^(2)x dx`

Answer

`(1)/(4)x^(2)-(x)/(4)sin2x-(1)/(8)cos 2x+C`

Explanation

`int xsin^(2)x dx`
`=int x{(1-cos2x)/(2)}dx=(1)/(2)intxdx-(1)/(2)int underset(I)(x)underset(II)(cos)2x dx`
`=(1)/(2)((x^(2))/(2))-(1)/(2)[x{intcos 2x dx}-int{(d)/(dx)(x)int cos2x}dx]`
`=(1)/(4)x^(2)-(1)/(2){(x)/(2)sin2x-int1xx(sin2x)/(2)dx}`
`=(x^(2))/(4)-(1)/(2){(x)/(2)sin 2x-(1)/(2)int sin2xdx}`
`=(x^(2))/(4)-(1)/(2){(x)/(2)sin 2x-(1)/(2)(-(1)/(2)cos2x)}+C`
`=(1)/(4)x^(2)-(x)/(4)sin2x-(1)/(8)cos 2x+C`