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Integrate with multiple variables in denominator?

  1. Mar 20, 2013 #1
    Hey everyone, I need to do the following integral. I just need a little help getting this started, I'm not sure where I need to go. Here is the problem:

    [itex]\int_{0}^{1-v} du \int_{0}^{\frac{1}{2}} dv \frac{1}{1+u^2-v^2} [/itex]

    I think I have the boundaries for the integral set up correctly, {0≤v≤1/2, 0≤u≤1-v}.
    I know that I will have to use [itex]\int dx \frac{1}{c^2+x^2} = \frac{1}{c}tan^ {-1}\frac{x}{2}[/itex]

    I began by trying to substitute 1-v^2 as a variable, but then I had to try to integrate tan^-1 with a bunch of square roots in it and that got really bad looking pretty quickly. Thoughts?
     
  2. jcsd
  3. Mar 20, 2013 #2

    haruspex

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    I would try a change of coordinates, like x = u+v, y = u-v. The region gets a little more complicated but it's not too bad. Don't forget the Jacobian.
     
  4. Mar 20, 2013 #3
    The original problem wants to change it from x and y to terms of u and v. I.e., x=u-v, y=u+v.
     
  5. Mar 20, 2013 #4

    Zondrina

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    Could you give us the original question with the original region?
     
  6. Mar 20, 2013 #5
    Sure, [itex]\int\int_{D}{} \frac{1}{1+xy} dxdy[/itex] D={(x,y);0≤x≤1,0≤y≤1}, x=u-v, y=u+v. That's all I am given.
     
  7. Mar 20, 2013 #6

    Zondrina

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    Follow what harup said and at least calculate your jacobian.

    ##dxdy = \frac{∂(x, y)}{∂(u, v)}dudv##
     
  8. Mar 20, 2013 #7

    haruspex

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    Ok, then you need to look at what you can substitute for v after the first integration. You have ##\frac1{\sqrt{1-v^2}} atan(\sqrt\frac{1-v}{{1+v}})##, right? Let ##\theta = atan(\sqrt\frac{1-v}{{1+v}})##, so ##tan^2(\theta) =\frac{1-v}{1+v}##. When you see tan-squared, what do you think of?
     
  9. Mar 20, 2013 #8

    haruspex

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    That integration range doesn't match what you have in u, v coordinates in the OP.
     
  10. Mar 20, 2013 #9
    [itex] \int_{v}^{1-v} [/itex]

    I think I had that part wrong.
     
  11. Mar 21, 2013 #10

    haruspex

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    It gets a bit messy. Something like v from -1/2 to +1/2, u from |v| to 1-|v|.
    Did you figure out what to do with the tan-squared?
     
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