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Nasty Integral - Help with Trig Substitution

  1. Jan 27, 2013 #1
    1. The problem statement, all variables and given/known data
    [itex]
    \int_{-p}^{p} \frac{2p}{(1+v^2)\sqrt{p^2 + v^2 +1 }} dv
    [/itex]


    2. Relevant equations
    [itex] 1 + \tan{\theta}^2 = \sec{\theta}^2 [/itex]

    3. The attempt at a solution
    I thought the best way to go about this was to rename some constants.
    Let [itex]\alpha^2 = 1 + p^2 [/itex] so that we are left with:
    [itex]
    \int_{-p}^{p} \frac{2p}{(1+v^2)\sqrt{\alpha^2+ v^2 }} dv
    [/itex]

    i then make the variable substitution:
    [itex] v = \alpha \tan{\theta} [/itex]
    [itex] dv = \alpha \sec{\theta}^2 d\theta [/itex]

    we now have:
    [itex]
    2p \int_{-p}^{p} \frac{1}{(1+\alpha^2 \tan^2{\theta})\sqrt{\alpha^2(1+\tan^2{\theta} )}} \alpha \sec{\theta}^2 d\theta
    [/itex]

    [itex]
    2p \int_{-p}^{p} \frac{1}{(1+\alpha^2 \tan^2{\theta})\alpha \sec{\theta}} \alpha \sec{\theta}^2 d\theta
    [/itex]

    [itex]
    2p \int_{-p}^{p} \frac{ \sec{\theta}}{(1+\alpha^2 \tan^2{\theta})} d\theta
    [/itex]

    at this point i hit a roadblock - the (1 - \alpha \tan{\theta} ) term seems to be an inextricable pain . other substitutions leave me with a similar problem.
    If anyone can point me in the direction of a better trig substitution, or perhaps an identity then your help would be greatly appreciated.
    thanks for reading
    -mrworf
     
  2. jcsd
  3. Jan 27, 2013 #2

    SammyS

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    Hello mrworf. Welcome to PF !

    First of all, [itex] \ \sec\theta^2\ [/itex] means [itex] \ \sec(\theta^2)\,\ [/itex] whereas [itex] \ \sec^2\theta\ [/itex] means [itex] \ (\sec\theta)^2\ .\ [/itex] I'm sure you want the latter for this problem.

    Also, you need to change the limits of integration to reflect the substitution you did.
     
  4. Jan 27, 2013 #3
    right on, with sammyS' suggestion, my equation becomes:

    [itex]2 p \int_{\tan^{-1}{\frac{p}{\alpha}}}^{\tan^{-1}{\frac{-p}{\alpha}}} \bigg[ \frac{\sec{\theta}}{(1+\alpha^2 \tan^2{\theta})} \bigg] d\theta [/itex]
     
  5. Jan 27, 2013 #4

    SammyS

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    I admit, it's still quite nasty !
     
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