1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Integration help!

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

    gif.latex?\int_{-\infty}^{0}&space;\frac{e^x}{1+cos^{2}(2x)}dx.gif

    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

    User Avatar
    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]
    so
    [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.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Integration help!
  1. Integral help (Replies: 4)

  2. Integration Help (Replies: 5)

  3. Help with an integral (Replies: 1)

  4. Integrals help (Replies: 7)

Loading...