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Homework Help: Simple integrals for gravitational potential

  1. Jan 7, 2013 #1
    1. The problem statement, all variables and given/known data
    2. Relevant equations

    I need help solving intergral…
    [tex]\int \frac{dx}{(a+x)^2}[/tex]

    3. The attempt at a solution
    I found the integral for…
    [tex] \int \frac{dx}{(a^2+x^2)} [/tex] = 1/a arctan x/a

    But I don’t know how to apply that to the original integral which is a little different

    [tex]\int \frac{dx}{(a+x)^2} = \int \frac{dx}{(a^2+x^2+2ax)}[/tex]

    I also need to solve the following integral
    [tex]\int \frac{dx}{(a+b-x)^2}[/tex]

    It’s not homework. The reason is that I want to work through the numbers that Rybczyk gives as equation 1 for gravitational potential in his paper “Gravitational Effect on Light Propagation”

    http://www.mrelativity.net/Gravitat...ravitational Effects on Light Propagation.htm

    I have simplified his equation in my post somewhat, as I already know how to separate the two terms separated by the minus sign, and assuming the gravitational constant G and the mass of the bodies are constant, I know that they can come out in front of the integral sign.

    Also, I changed the variable names to more familiar ones. I hope the variable name substitutions helps rather than hinders. If not I'll have to rewrite this whole post using Rybczyk's exact variables..

  2. jcsd
  3. Jan 7, 2013 #2


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    For the initial integral, use substitution, letting u = x+a .
  4. Jan 7, 2013 #3
    Thanks Sammy. It looks to me like du = dx in this case. Is that right?

    [tex]\int \frac{dx}{(a+x)^2} =[/tex]
    [tex]\int {(a+x)^{-2}}{dx}[/tex]

    u = a + x
    du = dx

    [tex]\int {(u)^{-2}}{du}=[/tex]
    [tex] -\frac{1}{u} + c[/tex]
    [edit] made correction (forgot the + c)

    Substituting a + x for u gives

    [tex] -\frac{1}{a+x} + c[/tex]

    Is that right?
    Last edited: Jan 7, 2013
  5. Jan 7, 2013 #4


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    Yes, it's right.
  6. Jan 7, 2013 #5
    Great, thanks Dick (quick reply!).

    Will try same for the second part of original question.

    NOTE: I editied post #3 to make a correction for the missing ( + c) for a constant.
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