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Homework Help: Integration help!

  1. Jan 11, 2014 #1
    1. The problem statement, all variables and given/known data
    Calculate the integral:


    2. Relevant equations


    3. The attempt at a solution

    I tried some trig identities, like t=tan(x/2). The cos^2 smells like an arctan derivative but I can't seem to think of anything that could work...
  2. jcsd
  3. Jan 13, 2014 #2
    You know cos^2(2x) is non-negative so it means 1+ cos^2(2x) is always positive
    Use the fact that 1+cos^2(2x) is bounded to evaluate this integral .
    Last edited: Jan 13, 2014
  4. Jan 14, 2014 #3
    Maybe write cos in exponents?
  5. Jan 14, 2014 #4

    Ray Vickson

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    Science Advisor
    Homework Helper

    I am beginning to doubt there is a simple closed-form solution. However, one can reduce it to a finite integration that might be preferable to use if you want an accurate numerical value. Call the integral J, and note that we can re-write it as an integral over [0,∞):
    [tex] J = \int_0^{\infty} f(x) \, dx, \:\: f(x) = \frac{e^{-x}}{1 + \cos^2(2x)} [/tex]
    Since ##\cos^2(2x)## is periodic with period ##\pi/2## we have
    [tex] f\left( n \frac{\pi}{2} + t \right) = \alpha^n f(t), \; \alpha = e^{-\pi/2} [/tex]
    [tex] J = \sum_{n=0}^{\infty} \alpha^n J_0 = \frac{J_0}{1-\alpha}, \text{ where }
    J_0 = \int_0^{\pi/2} f(x) \, dx. [/tex]
    For numerical work it might be better to work with J_0 instead of the original J.
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