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  3. 设f(x)连续，=∫0xf(x—t)costdt，求∫01f(x)dx．

设f(x)连续，=∫0xf(x—t)costdt，求∫01f(x)dx．

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问题 设f(x)连续，=∫0xf(x—t)costdt，求∫01f(x)dx．
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答案因为[*]所以 e2x=∫0xf(x—t)costdt[*]∫0xf(u)cos(x—u)du =cosx∫0xf(u)cosudu+sinx∫0xf(u)sinudu， 且2e2x=一sinx∫0xf(u)cosudu+f(x)cos2x+cosx∫0xf(u)sinudu+f(x)2sinx， 4e2x=f’(x)一cosx∫0xf(u)cosudu一f(x)sinxcosx—sinx∫0xf(u)sinudu+f(x)sinxcosx =f’(x)一e2x， 从而f’(x)=5e2x．于是，有 ∫01f(x)dx=xf(x)｜01一5∫01xe2xdx [*]
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