How to integrate this conundrum?

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Discussion Overview

The discussion revolves around the integration of the function x/(1+cos^2(x)) from 0 to π. Participants are exploring methods to prove the equality of this integral to π/(2√2), discussing various techniques and approaches to tackle the problem.

Discussion Character

  • Mathematical reasoning, Homework-related, Exploratory

Main Points Raised

  • One participant expresses uncertainty about integrating the function directly and suggests that a Taylor Expansion of cos^2(x) might be helpful.
  • Another participant proposes a substitution method, u=π/2-x, as a potential approach to simplify the integral.
  • A third participant questions the reasoning behind the substitution and asks for clarification on the technique being used.
  • A participant responds that the substitution is commonly useful for trigonometric integrals over the interval from 0 to π/2.

Areas of Agreement / Disagreement

Participants have not reached a consensus on the best method to approach the integral, and multiple strategies are being discussed without resolution.

I<3Gauss
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Does anyone know how to prove the following statement? I haven't messed with integrals for awhile and I have to say that I am kind of rusty on this. From initial attempts, it seems the integral on the left is not something you can integrate directly... Maybe Taylor Expansion of cos^2(x) would help?

∫x/(1+cos^2(x)) = pi/(2*√2) (integrated from pi to 0)

Thanks guys!
 
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So you're asking for [tex]\displaystyle\int\limits_0^\pi\left(\dfrac{x\cdot dx}{1+\cos^2\left(x\right)}\right)[/tex]? My first thought was substitute [tex]u=\dfrac\pi2-x[/tex] and doing something like averaging out the results. Anyone?

EDIT: GRRR ... why isn't there a \mathrm on here?
 
Whovian, What gave you the idea to make such a substitution? What technique are you using?
 
It's just a substitution that helps with a lot of trig integrals from 0 to pi/2.
 

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