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Integrating the sqrt of a function

  1. Mar 26, 2013 #1
    1. The problem statement, all variables and given/known data

    Apologies for the vague title, I'm not reall sure what I'm look at here! I am doing some revision on solving the Friedmann equations, and in a lot of cases I end up having to integrate a function that looks like

    (xan - 1)-1/2 da = dt

    where a is a function of t, x is a constant, and I am trying to find an equation that describes a as a function of t. The first examples I have found have n = -1, n = -2

    2. Relevant equations



    3. The attempt at a solution

    I do have a solution for one of these in my lecture notes. In the case where

    dt = (xa-1-1)-1/2

    the solution is given as

    t = x arctan ((xa-1-1)-1/2) - a1/2 (x - a)1/2

    I would normally look on wolfram and work through their step-by-step solution to teach myself the method, but even wolfram can't answer this one! Any help would be much appreciated.
     
    Last edited: Mar 26, 2013
  2. jcsd
  3. Mar 26, 2013 #2

    epenguin

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    You can check the stated result by differentiating which is often all the insight anyone needs.
     
  4. Mar 26, 2013 #3
    Thanks for the tip. I have tried that approach, and double-checked my differentiation with wolfram, but I can't get from the given solution to the starting point in that direction either!

    I started with the first term in the solution

    d/da x arctan ((xa-1-1)-1/2)

    . . . lots of substitutions . . .

    = x2(1+(xa-1-1)-1/2)-1(2a2(xa-1-1)3/2)-1

    Phew! And for the second term in the solution, I got

    d/da a1/2 (x - a)1/2 = (x-2a)(2(xa-a2)1/2)-1

    As I say, I have checked these differentiations with Wolfram so I am sure they are correct (although there is always the possibility of transcription errors when typing such complex formulae) but I can't see how they add up to give the original expression I was trying to integrate. What could I try next?
     
  5. Mar 27, 2013 #4

    epenguin

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    I don't have time today to go through that - perhaps another site member could. Showing more of your working wouldn't harm.

    Excercise for students!:approve:

    Are you sure you quoted result right? Some other bracket? For the second part I got

    - x a-1/2 (x - a)-1/2 .
     
    Last edited: Mar 27, 2013
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