integrate x*sinxdx

∫xsinx dx = x∫sinx - ∫d/dx(x)(∫sinxdx) dx .........(taking x as first function and sinx as second);

               = x(-cosx) - ∫d/dx(x) (-cosx) dx

               = -xcosx + ∫cosx dx

               = -xcosx + sinx + C

hence ∫xsinx dx = -xcosx + sinx + C

or you can write

∫xsinx dx = Im ∫x e^(ix) dx

               = Im (x∫e^(ix) dx - ∫d/dx(x) ∫e^(ix)dx dx)

               = Im (xe^(ix)/i - ∫e^(ix)/i dx)

               = Im (-ixe^(ix) + e^(ix)/i)

               = Im (-ixe^(ix) - ie^(ix))

               = Im (-ie^(ix) (x-1))

               = Im (-i(cosx + isinx)(x-1))

               = Im ((x-1)sinx -i(x-1)cosx)

               = (x-1)sinx