Prove $\int_{0}^{\pi}\frac{x^2}{\sqrt{5}-2\cos{x}}dx=\frac{\pi^3}{15}+2\pi\ln^2{\left(\frac{1+\sqrt{5}}{2}\right)}$ without contour integration

Asked Jan 10 '14 at 15:28

Active Apr 03 '14 at 22:40

Viewed 479 times

Show that $$\int_{0}^{\pi}\dfrac{x^2}{\sqrt{5}-2\cos{x}}dx=\dfrac{\pi^3}{15}+2\pi\ln^2{\left(\dfrac{1+\sqrt{5}}{2}\right)}$$

This thread demonstrates how contour integration can be used to solve the above integral, but I'm interested in finding alternative methods.

I believe one might utilize the Polylogarithm in some way. My attempt: I tried to rely on the identity $$2\cos{x}=e^{ix}+e^{-ix}.$$

Can someone think of a solution to this integral which doesn't involve the residue theorem?

Thank you

edited Apr 13 '17 at 12:20

Community

asked Jan 10 '14 at 15:28

math110

Just a random idea, but would the Taylor series of $\cos x$ be any help? – 2012ssohn Jan 10 '14 at 15:30
FWIW The integral has a singularity at $x=i\log\frac{1+\sqrt{5}}{2}$ so it's just a matter of finding the right contour... – Dan Piponi Jan 21 '14 at 03:06
@math110 Why do you think that you can find a solution without complex integration? – user37238 Jan 29 '14 at 08:54
5

@user37238: Feynman loved to tell people that he could do any integral that his MIT friends could do using contour integration techniques. I would just love to sick this one on him. – Ron Gordon Jan 29 '14 at 15:41
It was solved here – Felix Marin Feb 02 '14 at 13:58
2

This is not a duplicate, this question explicitly ask for a solution that didn't use residue. Unless someone produce a solution to the other question that didn't use residue, this should not be closed. – achille hui Feb 02 '14 at 14:22
@RonGordon It's deleted. Recently, I found you already solved while I was trying a slightly different approach. However, I think it will be sometimes deleted forever.Thanks. – Felix Marin Feb 03 '14 at 17:00
why closed it? Thank you for explain? – math110 Feb 03 '14 at 17:15
2

I evaluated the integral in the other thread without using contour integration. – Random Variable Feb 04 '14 at 12:31

0 Answers0