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Complex Analysis Integral Question

  1. Jul 30, 2015 #1


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    Gold Member

    1. The problem statement, all variables and given/known data

    Computer the integral:

    Integral from 0 to infinity of (d(theta)/(5+4sin(theta))

    2. Relevant equations
    integral 0 to 2pi (d(theta)/1+asin(theta)) = 2pi/(sqrt(1-a^2)) (-1<a<1)

    3. The attempt at a solution
    I've seen this integral be computed from 0 to 2pi, where the answer is 2pi/3 using the integral from 0 to 2pi (d(theta)/1+asin(theta)) = 2pi/(sqrt(1-a^2)) (-1<a<1)

    Is there a typo in the bounds? Should the bounds be from 0 to 2pi instead of 0 to infinity? :/ Or can this problem be solved with another trick/the same trick?
  2. jcsd
  3. Jul 30, 2015 #2


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    Homework Helper

    It looks like the bounds should be 0 to 2pi. Otherwise you have:
    ##\int_0^\infty \frac{1}{5+4\sin\theta} \, d\theta = \infty \int_0^{2\pi} \frac{1}{5+4\sin\theta} \, d\theta##
    since there are an infinite number of periods between 0 and 2pi.
    The function is always positive >1/9, so there is no practical application to integrating over infinity, since the integral will clearly be greater than ##\infty \frac19##.
  4. Jul 30, 2015 #3


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    Beautiful! As I suspected.
    Thank you for this.
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