The integration by parts rule looks like this:

∫ u * v' dx = u * v - ∫ ( v * u' ) dx

Hence in this example, we want to make our u = x and v' = sinx

So we now need to work out what u' and v are:

u' = 1 which is the easier of the two; to work out v, we should integrate v' = sinx, this will give us v = -cosx

Hence if we now subsititute these into the equations, we will find that:

∫ xsinx dx = -xcosx - ∫ (-cosx) dx

= -xcosx - (-sinx) + C (where C is the constant of integration)

= sinx - xcosx + C