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Appropriate Change of Variables for integration

  1. Feb 24, 2005 #1
    Can anyone give me any hints as to find a suitable change of variables for this integral.

    infinity
    /
    |dt/(a^2+t^2)^3/2 =
    |
    / -infinity


    =2/a^2 * integral below
    Pi/2
    /
    | cos t dt
    |
    / 0


    Thank you in advance
     
  2. jcsd
  3. Feb 24, 2005 #2
    [tex]\int_{-\infty}^{\infty}\frac{dt}{(a^2+t^2)^\frac{3}{2}} = \frac{2}{a^2} *
    \int_{0}^{\frac{\pi}{2}} \cos{t}dt [/tex]

    Is this correct?

    I think you can do a [itex]\tan^{-1}[/itex] substitution and use triangles to rewrite the integral.
     
    Last edited: Feb 24, 2005
  4. Feb 24, 2005 #3

    Hurkyl

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    Shouldn't it be the same change of variable as for

    [tex]
    \int \frac{dx}{a^2 + x^2}
    [/tex]

    or

    [tex]
    \int \sqrt{a^2 + x^2} \, dx
    [/tex]

    ?
     
  5. Feb 24, 2005 #4

    dextercioby

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    I've always supported hyperbolic trig.functions used in substitutions.In your case,it's ~[tex] \sinh x [/tex]...

    Daniel.
     
  6. Feb 24, 2005 #5
    Alright using a table of integrals and some algebra here is what I have so far:

    [tex]\int_{-\infty}^{\infty}\frac{dt}{(a^2+t^2)^\frac{3}{2}} =

    \int_{-\infty}^{\infty}\frac{a}{t^2(a^2+t^2)^\frac{3}{2}} + \frac{3}{t^2}}\int_{-\infty}^{\infty}\frac{dt}{(a^2+t^2)^\frac{1}{2}} [/tex]


    Am I getting anywhere...I don't think so...
     
  7. Feb 24, 2005 #6

    dextercioby

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    How did u pull that square 't' outta the integral...?:eek:

    Daniel.
     
  8. Feb 24, 2005 #7
    I used a table of integrals...and some simple algebra, unless I looked at the wrong intergral form, but I don't think I did, so anyway, where do I use the substitution?
     
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