第一个积分=∫(0到π/2)dy∫(0到π/2)sin(x+y)d(x+y)
=∫(0到π/2)[﹣cos(x+y)(x=π/2)+cos(x+y)(x=0)]dy
=∫(0到π/2)[﹣cos(y+π/2)+cosy]dy
=∫(0到π/2)[﹣cos(y+π/2)]dy+∫(0到π/2)cosydy
=﹣sin(y+π/2)(y=π/2)+sin(y+π/2)(y=0)+siny(y=π/2)-siny(y=0)
=2.
第二个积分=1/2∫(0到π/2)dy∫(0到π/2)xsin(x+y)d(x+y)
=1/2∫(0到π/2)dy∫(0到π/2)x×(-1)×dcos(x+y)
=1/2∫(0到π/2)dy[﹣xcos(x+y)(x=π/2)+xcos(x+y)(x=0)+∫(0到π/2)cos(x+y)dx]
=1/2∫(0到π/2)[﹣π/2×cos(y+π/2)+sin(x+y)(x=π/2)-sin(x+y)(x=0)]dy
=1/2∫(0到π/2)[﹣π/2×cos(y+π/2)+sin(y+π/2)-siny]dy
=﹣π/4×sin(y+π/2)(y=π/2)+π/4×sin(y+π/2)(y=0)-1/2×cos(y+π/2)(y=π/2)+1/2×cos(y+π/2)(y=0)+1/2×cosy(y=π/2)-1/2×cosy(y=0)
=π/4.
第三个积分=1/2∫(0到π/2)ydy∫(0到π/2)sin(x+y)dx (接下来的步骤与第一个积分类似)
第四个积分与第二个积分的方法类似.