How do I evaluate this integral? \(\displaystyle\lim_{{\epsilon\to{0}}}\int{\frac{{{1}}}{{{x}^{{{1}+\epsilon}}}}}{\left.{d}{x}\right.}\)

Blackettyl2j

Blackettyl2j

Answered question

2022-03-24

How do I evaluate this integral?
limϵ01x1+ϵdx

Answer & Explanation

Declan Cameron

Declan Cameron

Beginner2022-03-25Added 12 answers

If you consider the definite integral from 1 to x, then the limit is lnx:
1xdtt1+ϵ=[tϵϵ]1x
=1xϵϵ
=1eϵlnxϵ
=1(1ϵlnx+O(ϵ2))ϵlnx
as ϵ0

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