Joanna Benson

2022-01-04

I would like to show that

${\int}_{0}^{1}\frac{x-1}{\mathrm{ln}\left(x\right)}dx=\mathrm{ln}2$

Mary Nicholson

Beginner2022-01-05Added 38 answers

This is a classic example of differentiating inside the integral sign.

In particular, let

$J\left(\alpha \right)={\int}_{0}^{1}\frac{{x}^{\alpha}-1}{\mathrm{log}\left(x\right)}dx$

Then one has that

$\frac{d}{d\alpha}J\left(\alpha \right)={\int}_{0}^{1}\frac{d}{da}\frac{{x}^{\alpha}-1}{\mathrm{log}\left(x\right)}dx={\int}_{0}^{1}{x}^{\alpha}dx=\frac{1}{\alpha +1}$

and so we know that$J\left(\alpha \right)=\mathrm{log}(\alpha +1)+C$ . Noting that $J\left(0\right)=0$ tells us that $C=0$ and so $J\left(\alpha \right)=\mathrm{log}(\alpha +1)$

In particular, let

Then one has that

and so we know that

Jacob Homer

Beginner2022-01-06Added 41 answers

karton

Expert2022-01-11Added 613 answers

Making the substitution

since we recognize a Frullani integral type.

Find the local maximum and minimum values and saddle points of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function

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