1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Assistance with a very tough integral

  1. Dec 8, 2014 #1
    1. The problem statement, all variables and given/known data
    Evaluate the integral.

    2. Relevant equations

    For reference, this is the solution, but I do not know how to get here:

    $$\frac{a}{2}\ln{\frac{\xi+1}{\xi-1}} -\frac{x}{2}\ln{\frac{\xi+\frac{x}{a}}{\xi-\frac{x}{a}}}$$

    3. The attempt at a solution
    First step for me was to integrate by parts. I set $$c=a+\alpha$$ and integrated using c as my working variable.

    After integrating by parts once, I end up with:


    I am not sure what to do here. I was thinking of trying some sort of u substitution, maybe having $$u=\sqrt{\frac{c^2-a^2}{c^2-x^2}}$$.

    There are additional restrictions on how these all relate to each other. For instance:

    c > 0
    a > 0
    α > 0

    Perhaps they can assist me with further limiting the scope of this integral and making it evaluate. Right now, if I try to put this integral into Mathematica, I end up with a statement which includes the Appell Hypergeometric function.

    It should be noted that I also do not know what the limits of integration would be. Perhaps having the solution would give some insight into what the limits of integration are, but I do not see it. I have generated a stack of paper over the course of a week trying to figure out this integral, and I am getting absolutely nowhere. Some further assistance would be fantastic.

    As a side note: Please tell me if my questions are being somehow vague on these forums. I would greatly like to improve the clarity of my questions for others. Given that every thread I have created in the past several months on this forum has been completely devoid of replies, I have to wonder whether I am fundamentally missing some key piece of information in my descriptions that results in scaring at the potential help away.
  2. jcsd
  3. Dec 8, 2014 #2


    Staff: Mentor

    Have you tried working backward from the answer to see if you can see how to proceed?
  4. Dec 8, 2014 #3


    Staff: Mentor

    For this integral -- ##\int \frac{a}{c}\sqrt{\frac{c^2-x^2}{c^2-a^2}}dc## -- I'd be more inclined to try a trig substitution.
  5. Dec 10, 2014 #4
    Both of these are very good ideas. I will see if I can do these. I went to the professor and he said that it's probably in a table somewhere, so I do not think he did the algebra either. After seeing a square root of squares, I did default to thinking it must be some kind of triangle equality I could set up.
  6. Dec 10, 2014 #5


    Staff: Mentor

    Any time you have a sum or difference of squares, or the square root of a sum or difference of squares, trig substitution is a good strategy. Keep in mind here that x is kind of a red herring - the variable of integration is c.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted