2. 考研
  3. 设f(x)=∫0xcost/(1+sin2t)dt，求∫01f’(x)/(1+f2(x))dx．

设f(x)=∫0xcost/(1+sin2t)dt，求∫01f’(x)/(1+f2(x))dx．

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问题 设f(x)=∫0xcost/(1+sin2t)dt，求∫01f’(x)/(1+f2(x))dx．
选项
答案∫0π/2f’(x)/(1+f2(x))dx=∫0π/21/(1+f2(x))df(x)=arctanf(x)｜0π/2=arctanf(π/2)，因为f(π/2)=∫0π/2cost/(1+sin2t)dt=arctansint｜0π/2=π/4，所以原式=arctanπ/4．
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考研数学三

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