题目

∫1/x到1 [f(u)-f(1/x)]du 求导

提问时间：2020-12-22

答案

∵∫[f(u)-f(1/x)]du =∫f(u)du-∫f(1/x)du
=∫f(u)du-(1-1/x)f(1/x)
∴[∫[f(u)-f(1/x)]du]'=[∫f(u)du-(1-1/x)f(1/x)]'
=[∫f(u)du]'-[(1-1/x)f(1/x)]'
=-f(1/x)(1/x)'-[f(1/x)(1-1/x)'+(1-1/x)f'(1/x)(1/x)']
=f(1/x)/x²-f(1/x)/x²+(1-1/x)f'(1/x)/x²
=f'(1/x)(x-1)/x³.