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

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

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答案∫₀^π/2f’(x)/(1+f²(x))dx=∫₀^π/21/(1+f²(x))df(x)=arctanf(x)｜₀^π/2=arctanf(π/2)，因为f(π/2)=∫₀^π/2cost/(1+sin²t)dt=arctansint｜₀^π/2=π/4，所以原式=arctanπ/4．

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考研数学三

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