Question

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

Answer 1

n=0?

Answer 2

n...? :o

Answer 3

Nevermind, haha The equation wasn't showing up correctly. It is now, hang on a minute

Answer 4

Are you getting imaginary numbers with this?

Answer 5

Hmmm 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

Answer 6

I'm getting something crazy like -1/2 (l1 i)... I don't think that is right, merp ... sorry :(

Answer 7

when you think about it, all problems are tricky... otherwise they wouldn't be problems... :)

Answer 8

Hmm I found the problem on physicsforum.com. I might be able to get through this one if I just try hard enough c:

Answer 9

Good luck young grasshopper :)

Answer 10

http://www.physicsforums.com/showthread.php?t=610439

Hmmm trying to make sense of that...

Answer 11

Oh, I love physicsforums :) Did you get it?

Answer 12

Nah it's hurting by head XD maybe ill take a look at it later :3

Answer 13

\[\int\limits_{0}^{\infty}\frac{ \arctan (\pi x) -\arctan (x) }{ x } dx\]

\[=\int\limits_{0}^{\infty}\int\limits_{x}^{\pi x}\frac{1}{1+y^2}\frac{1}{x}dydx\]


\[=\int\limits_{0}^{\infty}\int\limits_{y/\pi}^{y}\frac{1}{1+y^2}\frac{1}{x}dxdy=\cdots\]

Answer 14

By the way, @zepdrix , can you speak Russian?

Answer 15

No sorry c:

Answer 16

Don'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

Answer 17

This 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

Answer 18

Google translating the text helped a little bit :D heh thanks

Answer 19

The second formula is your case, \(f(x)=\arctan x\).\[\int\limits_{0}^{\infty}\frac{ \arctan \pi x -\arctan x }{ x } dx=(0-\frac{\pi}2)\ln\frac1\pi=\frac\pi2\ln\pi\]

Answer 20

Ya 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

Answer 21

You can ask me anything, because russian mathematicians use sometimes another designation for popular objects in math.