2. 考研
  3. 设f(x)=∫0xarcsin(t-1)2dt，则∫01f(x)dx=________.

设f(x)=∫0xarcsin(t-1)2dt，则∫01f(x)dx=________.

admin2022-12-09  21
问题 设f(x)=∫0xarcsin(t-1)2dt，则∫01f(x)dx=________.
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答案[*]
解析 ∫01f(x)dx=xf(x)|01-∫01xarcsin(x-1)2dx
=f(1)-∫01[1+(x-1)]arcsin(x-1)2dx
=-∫01(x-1)arcsin(x-1)2d(x-1)
=-1/2∫01arcsin(x-1)2d(x-1)2=-1/2∫10arcsinxdx=1/2∫01arcsinxdx
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考研数学三

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