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  3. 设f(x)满足等式xf’(x)－f(x)＝，且f(1)＝4，则∫01f(x)dx＝______．

设f(x)满足等式xf’(x)－f(x)＝，且f(1)＝4，则∫01f(x)dx＝______．

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问题 设f(x)满足等式xf’(x)－f(x)＝，且f(1)＝4，则∫01f(x)dx＝______．
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答案[*]
解析 ∫01f(x)dx＝xf(x)｜01－∫01xf’(x)dx＝f(1)－∫01[f(x)＋
]dx
＝4－∫01f(x)dx＋∫01d(1－x)
＝4－∫01f(x)dx－∫01dt＝4－∫01f(x)dx－cos2θdθ
＝4－∫01f(x)dx－，
于是∫01f(x)dx＝2－．
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考研数学三

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