∫(0→1)dy∫(y^2→y)siny/y dx=∫(0→1)siny/y(y-y^2)dy 中的(y-y^2)是怎么来的?
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![∫(0→1)dy∫(y^2→y)siny/y dx=∫(0→1)siny/y(y-y^2)dy 中的(y-y^2)是怎么来的?](/uploads/image/z/8929651-67-1.jpg?t=%E2%88%AB%280%E2%86%921%29dy%E2%88%AB%28y%5E2%E2%86%92y%29siny%2Fy+dx%3D%E2%88%AB%280%E2%86%921%29siny%2Fy%28y-y%5E2%29dy+%E4%B8%AD%E7%9A%84%28y-y%5E2%29%E6%98%AF%E6%80%8E%E4%B9%88%E6%9D%A5%E7%9A%84%3F)
∫(0→1)dy∫(y^2→y)siny/y dx=∫(0→1)siny/y(y-y^2)dy 中的(y-y^2)是怎么来的?
∫(0→1)dy∫(y^2→y)siny/y dx=∫(0→1)siny/y(y-y^2)dy 中的(y-y^2)是怎么来的?
∫(0→1)dy∫(y^2→y)siny/y dx=∫(0→1)siny/y(y-y^2)dy 中的(y-y^2)是怎么来的?
∫(y^2→y)siny/y dx=[siny/y x]|(y^2→y)=(y-y^2)siny/y
这里是把siny/y看成常数来积分
∫(0→1)dy∫(y^2→y)siny/y dx
=∫(0→1)siny/ydy∫(y^2→y)dx
=∫(0→1)siny/ydy x|(y^2→y)
=∫(0→1)siny/y(y-y^2)dy
=∫(0→1)siny(1-y)dy
=∫(0→1)sinydy-∫(0→1)ysinydy
=-∫(0→1)dcosy+∫(0→1)ydcosy
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∫(0→1)dy∫(y^2→y)siny/y dx
=∫(0→1)siny/ydy∫(y^2→y)dx
=∫(0→1)siny/ydy x|(y^2→y)
=∫(0→1)siny/y(y-y^2)dy
=∫(0→1)siny(1-y)dy
=∫(0→1)sinydy-∫(0→1)ysinydy
=-∫(0→1)dcosy+∫(0→1)ydcosy
=-cosy|(0→1) +ycosy|(0→1)-∫(0→1)cosydy
=-(cos1-cos0)+(1cos1-0cos0)-∫(0→1)dsiny
=-cos1+1+cos1-0-siny|(0→1)
=1-(sin1-sin0)
=1-sin1
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