设f(x)在[0,+∞)上可导,f(0)=0,且f(x)的反函数为g(x),若∫0f(x)g(t)dt=∫0xtarcsin(t-1)2dt.求∫01f(x)dx.

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问题 设f(x)在[0,+∞)上可导,f(0)=0,且f(x)的反函数为g(x),若∫0f(x)g(t)dt=∫0xtarcsin(t-1)2dt.求∫01f(x)dx.

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答案已知等式两边同时对x求导,得 g[f(x)]f’(x)=xarcsin(x-1)2. 又因为g[f(x)]=x,所以当x≠0时,有 f’(x)=arcsin(x-1)2, f(1)=f(0)+∫01f’(x)dx=∫01arcsin(x-1)2dx. 故 I=∫01f(x)dx=xf(x)|01-∫01xf’(x)dx=f(1)-∫01xarcsin(x-1)2dx =∫01arcsin(x-1)2dx-∫01xarcsin(x-1)2dx =∫01(1-x)arcsin(x-1)2dx[*]-∫-10tarcsin t2dt [*]

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