设f’(x)=arcsin(x一1)2及f(0)=0,求∫01f(x)dx.

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问题 设f’(x)=arcsin(x一1)2及f(0)=0,求∫01f(x)dx.

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答案01 f(x)dx=∫01 f(x)d(x-1)=(x-1)f(x)|01-∫01 f’ (x)(x-1)dx =—∫01 (x-1)arcsin(x-1) 2dx[*]

解析 若已知f(a)=0或f(b)=0时,则在分部积分时可用下面的小技巧简化计算.
    若已知f(a)=0,则∫abf(x)dx=∫abf(x)d(x一b)=一∫ab(x一b)f’(x)dx.
    若已知f(b)=0,则∫abf(x)dx=∫abf(x)d(x一a)=一∫ab(x一a)f’(x)dx.
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