这几个简单的 积分如何求得~这几个积分如何求得~
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![这几个简单的 积分如何求得~这几个积分如何求得~](/uploads/image/z/13813600-40-0.jpg?t=%E8%BF%99%E5%87%A0%E4%B8%AA%E7%AE%80%E5%8D%95%E7%9A%84+%E7%A7%AF%E5%88%86%E5%A6%82%E4%BD%95%E6%B1%82%E5%BE%97%7E%E8%BF%99%E5%87%A0%E4%B8%AA%E7%A7%AF%E5%88%86%E5%A6%82%E4%BD%95%E6%B1%82%E5%BE%97%7E)
这几个简单的 积分如何求得~这几个积分如何求得~
这几个简单的 积分如何求得~
这几个积分如何求得~
这几个简单的 积分如何求得~这几个积分如何求得~
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.