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experimentX
Group Title
Evaluate:
\[ \displaystyle \int_0^{\infty} \dfrac{(\log x)^2}{x^2 + 1} dx \]
 2 years ago
 2 years ago
experimentX Group Title
Evaluate: \[ \displaystyle \int_0^{\infty} \dfrac{(\log x)^2}{x^2 + 1} dx \]
 2 years ago
 2 years ago

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experimentX Group TitleBest ResponseYou've already chosen the best response.0
the problem hasn't been solved http://math.stackexchange.com/questions/191736/helpwithintegratingdisplaystyleint0inftydfraclogx2x21
 2 years ago

mukushla Group TitleBest ResponseYou've already chosen the best response.0
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\[\int_{\infty}^{0} \dfrac{t^2e^t}{1+ e^{2t}} \text{d}t=\int_{\infty}^{0} t^2e^t \sum_{n=0}^{\infty}(1)^ne^{2nt} \text{d}t= \sum_{n=0}^{\infty} (1)^n\int_{\infty}^{0} t^2e^te^{2nt} \text{d}t\\=\sum_{n=0}^{\infty} (1)^n \frac{2}{(2n+1)^3}\]
 2 years ago

mukushla Group TitleBest ResponseYou've already chosen the best response.0
but i'd like to know how can we go with complex integration
 2 years ago

experimentX Group TitleBest ResponseYou've already chosen the best response.0
the series looks like Fourier expansion.
 2 years ago

experimentX Group TitleBest ResponseYou've already chosen the best response.0
hold on .. is that 3?
 2 years ago

mukushla Group TitleBest ResponseYou've already chosen the best response.0
lol .. yes
 2 years ago

mukushla Group TitleBest ResponseYou've already chosen the best response.0
this one is reachable by fourier series i think
 2 years ago

experimentX Group TitleBest ResponseYou've already chosen the best response.0
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}]
 2 years ago

experimentX Group TitleBest ResponseYou've already chosen the best response.0
but this seems nice http://www.wolframalpha.com/input/?i=sum[%281%29^n%2F%282n%2B1%29^3%2C+{n%2C+0%2C+Infinity}]
 2 years ago

mukushla Group TitleBest ResponseYou've already chosen the best response.0
i made it worser
 2 years ago

experimentX Group TitleBest ResponseYou've already chosen the best response.0
No not really ,,, this is nice ... and interesting http://www.wolframalpha.com/input/?i=sum[%281%29^n%2F%282n%2B1%29^3%2C+{n%2C+1%2C+Infnity}]
 2 years ago

mukushla Group TitleBest ResponseYou've already chosen the best response.0
ahh yeah so this is reachable man
 2 years ago

experimentX Group TitleBest ResponseYou've already chosen the best response.0
probably some nasty Fourier analysis.
 2 years ago

eliassaab Group TitleBest ResponseYou've already chosen the best response.1
Hae you tried using the residue theorem?
 2 years ago

experimentX Group TitleBest ResponseYou've already chosen the best response.0
tried ... but stuck. I need picture of contour .... that t^2 term is bugging me badly http://openstudy.com/users/experimentx#/updates/50476f64e4b0c3bb09860ba6
 2 years ago

eliassaab Group TitleBest ResponseYou've already chosen the best response.1
See http://en.wikipedia.org/wiki/Methods_of_contour_integration Example (V)
 2 years ago

experimentX Group TitleBest ResponseYou've already chosen the best response.0
thanks ... i this is helpful ...
 2 years ago

experimentX Group TitleBest ResponseYou've already chosen the best response.0
i'll try it ... if it get answer i'll post solution.
 2 years ago

mukushla Group TitleBest ResponseYou've already chosen the best response.0
dw:1347643651001:dw
 2 years ago

mukushla Group TitleBest ResponseYou've already chosen the best response.0
man this contour works
 2 years ago

experimentX Group TitleBest ResponseYou've already chosen the best response.0
did you try it?
 2 years ago

experimentX Group TitleBest ResponseYou've already chosen the best response.0
oh great ... but I have QM exam six days later.
 2 years ago

mukushla Group TitleBest ResponseYou've already chosen the best response.0
\[\oint \frac{z^2 e^z}{1+e^{2z}} dz=a_{1}\]
 2 years ago

mukushla Group TitleBest ResponseYou've already chosen the best response.0
man try it after exams
 2 years ago

experimentX Group TitleBest ResponseYou've already chosen the best response.0
sure ... 14 days to go.
 2 years ago
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