definite integral

Math Challenge Problem:

$\displaystyle \int^{2}_{1} \frac{\ln(x-1)}{x} \ dx $

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Originally Posted by Random Variable

Math Challenge Problem:

$\displaystyle \int^{2}_{1} \frac{\ln(x-1)}{x} \ dx $

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Let's denote:

$\displaystyle I=\displaystyle\int^{2}_{1} \dfrac{\ln(x-1)}{x} \ dx $

with the substitution:

$\displaystyle t=\ln (x-1)$ we obtain:

$\displaystyle I=\displaystyle\int^{0}_{-\infty} \dfrac{te^t\;dt}{e^t+1} $

Now, apply twice integration by parts.

Fernando Revilla

$\displaystyle \int \frac{t e^{t}}{e^{t}+1} \ dt $ does not have an antiderivative that can be expressed as a finite number of elementary functions. So it's not quite that straightforward.

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Originally Posted by Random Variable

Math Challenge Problem:

$\displaystyle \int^{2}_{1} \frac{\ln(x-1)}{x} \ dx $

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Setting $\displaystyle x-1= u$ the integral becomes...

$\displaystyle \displaystyle \int_{0}^{1} \frac{\ln u}{1+u}\ du$ (1)

Now if You consider that is...

$\displaystyle \displaystyle \frac{1}{1+u} = \sum_{n=0}^{\infty} (-1)^{n}\ u^{n}$ (2)

... and...

$\displaystyle \displaystyle \int_{0}^{1} u^{n}\ \ln u = - \frac{1}{(n+1)^{2}}$ (3)

... You obtain...

$\displaystyle \displaystyle \int_{0}^{1} \frac{\ln u}{1+u}\ du = \sum_{n=1}^{\infty} \frac{(-1)^{n}}{n^{2}} = - \frac{\pi ^{2}}{12}$ (4)

http://digilander.libero.it/luposaba...ato[1].jpg

Merry Christmas from Italy

$\displaystyle \chi$ $\displaystyle \sigma$

Quote:

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Originally Posted by Random Variable

$\displaystyle \int \frac{t e^{t}}{e^{t}+1} \ dt $ does not have an antiderivative that can be expressed as a finite number of elementary functions. So it's not quite that straightforward.

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So what?. You can find $\displaystyle I$ (definite integral) by means of an equation.

Fernando Revilla

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Originally Posted by chisigma

Setting $\displaystyle x-1= u$ the integral becomes...
$\displaystyle \displaystyle \int_{0}^{1} \frac{\ln u}{1+u}\ du = \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^{2}} = \frac{\pi ^{2}}{12}$ (4)

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Mathematica and WolframAlpha both give the negative of your answer, so I'm guessing you have a sign error in there somewhere. Not sure where, though.

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Originally Posted by Ackbeet

Mathematica and WolframAlpha both give the negative of your answer, so I'm guessing you have a sign error in there somewhere. Not sure where, though.

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I think he just left out the negative by mistake.

Some years ago I 'found' the following formula...

$\displaystyle \displaystyle \int_{0}^{1} u^{n}\ \ln^{m} u\ du = (-1)^{m}\ \frac{m!}{(n+1)^{m+1}}$ (1)

In that case is $\displaystyle m=1$ so that...

$\displaystyle \displaystyle \int_{0}^{1} u^{n}\ \ln u\ du = - \frac{1}{(n+1)^{2}}$ (2)

Thank a lot to Ackbeet!...

http://digilander.libero.it/luposaba...ato[1].jpg

Merry Christmas from Italy

$\displaystyle \chi$ $\displaystyle \sigma$

My solution:

Let $\displaystyle x = 1+e^{-t} $

$\displaystyle = -\int^{0}_{\infty} \frac{\ln(e^{-t})}{1+e^{-t}} \ e^{-t} \ dt $

$\displaystyle = - \int^{\infty}_{0} \frac{te^{-t}}{1+e^{-t}} \ dt $

$\displaystyle = \int^{\infty}_{0} t \sum^{\infty}_{k=1} (-1)^{k} e^{-kt} \ dt $ (since $\displaystyle e^{-t} < 1 $ for $\displaystyle 0<t<\infty $ )

now integrate by parts

$\displaystyle t \sum_{k=1}^{\infty} (-1)^{k+1} \frac{e^{-kt}}{k} \Big|^{\infty}_{0} - \int^{\infty}_{0} \sum^{\infty}_{k=1} (-1)^{k+1} \ \frac{e^{-kt}}{k} \ dt $

$\displaystyle 0 + \int^{\infty}_{0} \sum^{\infty}_{k=1} (-1)^{k} \ \frac{e^{-kt}}{k} $

$\displaystyle = \sum^{\infty}_{k=1} \int^{\infty}_{0} (-1)^{k} \ \frac{e^{-kt}}{k}
$ (assuming$\displaystyle f_{k}(t) = (-1)^{k} \ \frac{e^{-kt}}{k} $ converges uniformly on $\displaystyle 0 \le t<\infty $)

$\displaystyle = -\sum^{\infty}_{k=1} (-1)^{k+1} \ \frac{1}{k^{2}} $

$\displaystyle = \sum^{\infty}_{k=1} (-1)^{k} \ \frac{1}{k^{2}} = -\frac{\pi^{2}}{12} $

Quote:

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Originally Posted by FernandoRevilla

So what?. You can find $\displaystyle I$ (definite integral) by means of an equation.

Fernando Revilla

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How do you expect to get $\displaystyle \pi^2$ in there? I'm not saying you're wrong, but I really don't see how you can find $\displaystyle I$ by integration by parts or by "means of an equation" - perhaps you can be more precise and provide your solution?

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Originally Posted by Bruno J.

I'm not saying you're wrong,

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You can say it. :)

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but I really don't see how you can find $\displaystyle I$ by integration by parts or by "means of an equation" - perhaps you can be more precise and provide your solution?

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I can't. I had an stupid computational mistake. Sorry. :)

Fernando Revilla

I'm able to show that:

$\displaystyle \begin{aligned} \int_{0}^{1}\frac{\ln\left(x+1\right)}{x}\;{dt} &=\int_{0}^{1}\bigg(\sum_{k = 0}^{\infty}\frac{(-x)^k}{k+1}\bigg) \;{dx} = \displaystyle \sum_{k=0}^{\infty}\bigg(\int_{0}^{1}\frac{(-x)^k}{k+1} \;{dx}\bigg)\\ &= \sum_{k=0}^{\infty}\bigg[\frac{(-1)^{k}x^{k+1}}{(k+1)^2}\;{dx}\bigg]_{x = 0}^{x = 1} = \sum_{k=0}^{\infty}\frac{(-1)^{k}}{(k+1)^2} = \frac{\pi^2}{12}.\end{aligned}$

But I'm lacking the change of variable to show this:

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

What 'substitution' could do the trick? Mental block!