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zepdrix
 3 years ago
Calc 3 Problem  Tricky Integral
\[\int\limits_{0}^{\infty}\frac{ \arctan \pi x \arctan x }{ x } dx\]
zepdrix
 3 years ago
Calc 3 Problem  Tricky Integral \[\int\limits_{0}^{\infty}\frac{ \arctan \pi x \arctan x }{ x } dx\]

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johnsonshelby
 3 years ago
Best ResponseYou've already chosen the best response.0Nevermind, haha The equation wasn't showing up correctly. It is now, hang on a minute

johnsonshelby
 3 years ago
Best ResponseYou've already chosen the best response.0Are you getting imaginary numbers with this?

zepdrix
 3 years ago
Best ResponseYou've already chosen the best response.0Hmmm I don't think so :D And for the life of me, I can't remember the hint that the teacher gave us lol. Something aboutttttt try to rewrite it as the derivative of something else that is easier to integrate, orrrrrr maybe, rewrite it as a double integral somehow, blah i forget what he said XD lolol

johnsonshelby
 3 years ago
Best ResponseYou've already chosen the best response.0I'm getting something crazy like 1/2 (l1 i)... I don't think that is right, merp ... sorry :(

dpaInc
 3 years ago
Best ResponseYou've already chosen the best response.0when you think about it, all problems are tricky... otherwise they wouldn't be problems... :)

zepdrix
 3 years ago
Best ResponseYou've already chosen the best response.0Hmm I found the problem on physicsforum.com. I might be able to get through this one if I just try hard enough c:

johnsonshelby
 3 years ago
Best ResponseYou've already chosen the best response.0Good luck young grasshopper :)

zepdrix
 3 years ago
Best ResponseYou've already chosen the best response.0http://www.physicsforums.com/showthread.php?t=610439 Hmmm trying to make sense of that...

johnsonshelby
 3 years ago
Best ResponseYou've already chosen the best response.0Oh, I love physicsforums :) Did you get it?

zepdrix
 3 years ago
Best ResponseYou've already chosen the best response.0Nah it's hurting by head XD maybe ill take a look at it later :3

Zarkon
 3 years ago
Best ResponseYou've already chosen the best response.1\[\int\limits_{0}^{\infty}\frac{ \arctan (\pi x) \arctan (x) }{ x } dx\] \[=\int\limits_{0}^{\infty}\int\limits_{x}^{\pi x}\frac{1}{1+y^2}\frac{1}{x}dydx\] \[=\int\limits_{0}^{\infty}\int\limits_{y/\pi}^{y}\frac{1}{1+y^2}\frac{1}{x}dxdy=\cdots\]

klimenkov
 3 years ago
Best ResponseYou've already chosen the best response.0By the way, @zepdrix , can you speak Russian?

klimenkov
 3 years ago
Best ResponseYou've already chosen the best response.0Don't read the text, just see the formula. http://ru.wikipedia.org/wiki/%D0%A4%D0%BE%D1%80%D0%BC%D1%83%D0%BB%D1%8B_%D0%A4%D1%80%D1%83%D0%BB%D0%BB%D0%B0%D0%BD%D0%B8

zepdrix
 3 years ago
Best ResponseYou've already chosen the best response.0This problem is starting to make sense, I'm trying to understand how Zarkon switched the integrals though. From the second to the third step, we are integrating with resepect to X first now, but how did he come up with the limits? :( hmm

zepdrix
 3 years ago
Best ResponseYou've already chosen the best response.0Google translating the text helped a little bit :D heh thanks

klimenkov
 3 years ago
Best ResponseYou've already chosen the best response.0The second formula is your case, \(f(x)=\arctan x\).\[\int\limits_{0}^{\infty}\frac{ \arctan \pi x \arctan x }{ x } dx=(0\frac{\pi}2)\ln\frac1\pi=\frac\pi2\ln\pi\]

zepdrix
 3 years ago
Best ResponseYou've already chosen the best response.0Ya i need to try and show the steps though :d not just a shortcut. And the proof on that page is rather involved :3 heh

klimenkov
 3 years ago
Best ResponseYou've already chosen the best response.0You can ask me anything, because russian mathematicians use sometimes another designation for popular objects in math.

Zarkon
 3 years ago
Best ResponseYou've already chosen the best response.1\[\int\limits_{0}^{\infty}\frac{1}{1+y^2}dy=\lim_{t\to\infty}\int\limits_{0}^{t}\frac{1}{1+y^2}dy\] \[=\left.\lim_{t\to\infty}\arctan(y)\right _{0}^{t}=\lim_{t\to\infty}\arctan(t)\arctan(0)=\frac{\pi}{2}0=\frac{\pi}{2}\] combine with the above
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