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Trig substitution help

  1. Oct 21, 2009 #1
    1. The problem statement, all variables and given/known data

    I want to integrate:

    [tex]\int^y_0\frac{y-x}{(a^2(y)-a^2(x))^b}dx[/tex]

    2. Relevant equations

    a2(y) means that a is a function of y. similarly for [a(x)]2. so [a(x)]2 is a functions that depends on x.

    3. The attempt at a solution

    I tried integration by parts:
    let
    u=y-x so u'=-1
    [tex]v'=(a^2(y)-a^2(x))^{-b}[/tex]
    and now i am struggling to integrate v'.
     
  2. jcsd
  3. Oct 21, 2009 #2
    Re: integration

    Is this the whole problem? Or are you, perhaps, supposed to do an additional computation with this integral? I ask because (since you don't know what function a is) you won't get any good expression for this integral by itself.
     
  4. Oct 21, 2009 #3

    Mark44

    Staff: Mentor

    Re: integration

    I think I would start with a trig substitution first. Rather than remembering which formulas go with which situations, I draw a right triangle and label the sides and hypotenuse. I would label the hypotenuse a(y), and either of the other two sides as a(x). That leaves sqrt(a(y)^2 - a(x)^2) for the other side.
     
  5. Oct 21, 2009 #4
    Re: integration

    this is the whole problem. the answer should be interms of the function a.

    why would drawing a triangle help with choosing a substitution?
     
  6. Oct 21, 2009 #5

    Mark44

    Staff: Mentor

    Re: integration

    Drawing a picture of a triangle is helpful when you're doing trig substitution, because it helps you establish the relationships between your substitution variable and the variables in your problem. You might be able to get by without drawing a picture, but you're probably more prone to making a mistake.
     
  7. Oct 21, 2009 #6
    Re: integration

    i see what you mean.

    so how about the substitution: a(x)=a(y)cos(x)
    ?
     
  8. Oct 21, 2009 #7

    Mark44

    Staff: Mentor

    Re: integration

    Yes, that works. If you are using the triangle drawing, that would correspond to labelling the horizontal leg a(x) and the hypotenuse a(y).
     
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