求高数二重积分详细题解谢谢
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∫∫[D] x*cos(x+y)*dσ
=∫[0,π] dx∫[0,x] x*cos(x+y)*dy
=∫[0,π] xdx {sin(x+y)|[0,x]}
=∫[0,π] [xsin2x-xsinx]dx
=-1/2∫[0,π] xdcos2x +∫[0,π] xdcosx
=-1/2∫[0,π] xdcos2x +∫[0,π] xdcosx
={-1/2[xcos2x-(xsin2x)/2] +xcosx-sinx} |[0,π]
=-3π/2
=∫[0,π] dx∫[0,x] x*cos(x+y)*dy
=∫[0,π] xdx {sin(x+y)|[0,x]}
=∫[0,π] [xsin2x-xsinx]dx
=-1/2∫[0,π] xdcos2x +∫[0,π] xdcosx
=-1/2∫[0,π] xdcos2x +∫[0,π] xdcosx
={-1/2[xcos2x-(xsin2x)/2] +xcosx-sinx} |[0,π]
=-3π/2
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