experimentX
  • experimentX
Evaluate: \[ \displaystyle \int_0^{\infty} \dfrac{(\log x)^2}{x^2 + 1} dx \]
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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chestercat
  • chestercat
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experimentX
  • experimentX
http://tinyurl.com/9csdea3
experimentX
  • experimentX
the problem hasn't been solved http://math.stackexchange.com/questions/191736/help-with-integrating-displaystyle-int-0-infty-dfrac-log-x2x2-1
anonymous
  • anonymous
*

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anonymous
  • anonymous
going with \[t=\log x\]\[\displaystyle \int_{-\infty}^{\infty} \dfrac{t^2e^t}{1+ e^{2t}} \text{d}t=\int_{-\infty}^{0} \dfrac{t^2e^t}{1+ e^{2t}} \text{d}t+\int_{0}^{\infty} \dfrac{t^2e^t}{1+ e^{2t}} \text{d}t\]first one for example\[\int_{-\infty}^{0} \dfrac{t^2e^t}{1+ e^{2t}} \text{d}t=\int_{-\infty}^{0} t^2e^t \sum_{n=0}^{\infty}(-1)^ne^{2nt} \text{d}t= \sum_{n=0}^{\infty} (-1)^n\int_{-\infty}^{0} t^2e^te^{2nt} \text{d}t\\=\sum_{n=0}^{\infty} (-1)^n \frac{2}{(2n+1)^3}\]
anonymous
  • anonymous
but i'd like to know how can we go with complex integration
experimentX
  • experimentX
the series looks like Fourier expansion.
anonymous
  • anonymous
yeah
experimentX
  • experimentX
hold on .. is that 3?
anonymous
  • anonymous
lol .. yes
anonymous
  • anonymous
this one is reachable by fourier series i think
experimentX
  • experimentX
you made things a bit more complicated http://www.wolframalpha.com/input/?i=sum[1%2F%282n%2B1%29^3%2C+{n%2C+0%2C+Infinity}]
experimentX
  • experimentX
but this seems nice http://www.wolframalpha.com/input/?i=sum[%28-1%29^n%2F%282n%2B1%29^3%2C+{n%2C+0%2C+Infinity}]
anonymous
  • anonymous
i made it worser
experimentX
  • experimentX
No not really ,,, this is nice ... and interesting http://www.wolframalpha.com/input/?i=sum[%28-1%29^n%2F%282n%2B1%29^3%2C+{n%2C+1%2C+Infnity}]
anonymous
  • anonymous
ahh yeah so this is reachable man
experimentX
  • experimentX
probably some nasty Fourier analysis.
anonymous
  • anonymous
*
anonymous
  • anonymous
Hae you tried using the residue theorem?
experimentX
  • experimentX
tried ... but stuck. I need picture of contour .... that t^2 term is bugging me badly http://openstudy.com/users/experimentx#/updates/50476f64e4b0c3bb09860ba6
anonymous
  • anonymous
See http://en.wikipedia.org/wiki/Methods_of_contour_integration Example (V)
experimentX
  • experimentX
thanks ... i this is helpful ...
experimentX
  • experimentX
*think
experimentX
  • experimentX
i'll try it ... if it get answer i'll post solution.
anonymous
  • anonymous
|dw:1347643651001:dw|
anonymous
  • anonymous
man this contour works
experimentX
  • experimentX
did you try it?
anonymous
  • anonymous
yes
experimentX
  • experimentX
oh great ... but I have QM exam six days later.
anonymous
  • anonymous
\[\oint \frac{z^2 e^z}{1+e^{2z}} dz=a_{-1}\]
anonymous
  • anonymous
man try it after exams
experimentX
  • experimentX
sure ... 14 days to go.

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