I would like to show that \int_0^1\frac{x-1}{\ln(x)}dx=\ln 2

Joanna Benson

Joanna Benson

Answered question

2022-01-04

I would like to show that
01x1ln(x)dx=ln2

Answer & Explanation

Mary Nicholson

Mary Nicholson

Beginner2022-01-05Added 38 answers

This is a classic example of differentiating inside the integral sign.
In particular, let
J(α)=01xα1log(x)dx
Then one has that
ddαJ(α)=01ddaxα1log(x)dx=01xαdx=1α+1
and so we know that J(α)=log(α+1)+C. Noting that J(0)=0 tells us that C=0 and so J(α)=log(α+1)
Jacob Homer

Jacob Homer

Beginner2022-01-06Added 41 answers

01x1logxdx=0101xtdtdx
=01xtdxdt
=0111+tdt
=log(2)
karton

karton

Expert2022-01-11Added 613 answers

Making the substitution u=lnx, we get
I=0eu1ueudu=0+e2sessds=ln21=ln2
since we recognize a Frullani integral type.

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