二重積分x*cos(x+y),其中D是頂點分別為(0,0),(π,0),(π,π)圍成的三角形區(qū)域.
計算二重積分xcos(x+y)dσ ,其中D是頂點分別為(0,0),(π,0)和(π,π)的三角形閉區(qū)域.
∬xcos(x+y)dxdy
=[0,π]∫xdx∫[0,x]cos(x+y)d(x+y)
=[0,π]∫xdx[sin(x+y)]︱[0,x]
①式 =[0,π]∫x(sin2x-sinx)dx
=[0,π][∫xsin2xdx-∫xsinxdx]
=[0,π][-(1/2)∫xd(cos2x)+∫xd(cosx)]
②式 =[0,π]{-(1/2)[xcos2x-∫cos2xdx]+[xcosx-∫cosxdx]}
=[0,π]{-(1/2)[xcos2x-(1/2)sin2x]+[xcosx-sinx]}
=[0,π]{-(1/2)xcos2x+(1/4)sin2x+xcosx-sinx}
=-(1/2)π-π
=-(3/2)π
其中一式和2式都是怎么來的?