设f(x)=∫0xecostdt,求∫0πf(x)cosxdx.

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问题 设f(x)=∫0xecostdt,求∫0πf(x)cosxdx.

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答案0πf(x)cosx dx=∫0πf(x)d(sinx)=f(x)sinx|0π—∫0πf’(x)sinx dx =-∫0π’(x)sinx dx=一∫0πecosxsinx dx=∫0πecosxd(cosx) =ecosx|0π=e-1-e.

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