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  3. 设y’=arctan(x-1)2，y(0)=0，求∫01y(x)dx．

设y’=arctan(x-1)2，y(0)=0，求∫01y(x)dx．

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问题 设y’=arctan(x-1)2，y(0)=0，求∫01y(x)dx．
选项
答案∫01y(x)dx=xy(x)｜01-∫01xarctan(x-1)2dx=y(1)-01(x-1)arctan(x-1)2d(x-1)-∫01arctan(x-1)2dx=-1/2∫01arctan(x-1)2d(x-1)2=1/2∫01arctantdt=1/2(tarctant｜01-∫01t/(1+t2)dt)=π/8-1/4ln2．
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考研数学三

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