Proving Integral of an Irrational Function

  1. Hey,

    1. The problem statement, all variables and given/known data
    (From an Integration Table)
    Prove,
    [tex]
    {\int}{\sqrt {{{a}^{2}} - {{x}^{2}}}}{dx} = {{\frac {1}{2}}{\left(}{{{x}{\sqrt {{{a}^{2}} - {{x}^{2}}}}} + {{{a}^{2}}{\arcsin{\frac {x}{a}}}}}{\right)}}{,}{\,}{\,}{\,}{\,}{\,}{\,}{{{|}{x}{|}}{\leq}{{|}{a}{|}}}
    [/tex]

    2. Relevant equations
    Knowledge of Trigonometric Substitution for Integration ("Backwards" and "Forwards").

    Knowledge of integration techniques involving the form,
    [tex]
    {\sqrt {{{a}^{2}} - {{x}^{2}}}}
    [/tex]

    Integration by Parts (IBP),
    [tex]
    {\int}{d{\left[}{u(x)}{v(x)}{\right]}} = {\int}{\biggl[}{u(x)}{d[v(x)]} + {v(x)}{d[u(x)]}{\biggl]}
    [/tex]

    [tex]
    {\int}{u(x)}{d[v(x)]} = {{u(x)}{v(x)}} - {\int}{v(x)}{d[u(x)]}
    [/tex]

    3. The attempt at a solution
    What is bothering me about this integral is that I do not have a [itex]{x}[/itex] term on the outside of the radical which is preventing from evaluating this integral by normal convention.

    Let,
    [tex]
    {I} = {\int}{\sqrt {{{a}^{2}} - {{x}^{2}}}}{dx}
    [/tex]

    In applying trigonometric substitution - consider the right triangle,
    [​IMG]

    So,
    [tex]
    {{a}{\sin{\theta}}} = {\sqrt {{{a}^{2}} - {{x}^{2}}}}
    [/tex]

    Solving by "forward" substitution,
    [tex]
    {x} = {\pm}{\sqrt {{{a}^{2}}{{\cos}^{2}{\theta}}}}
    [/tex]

    However, how would you justify which root to take: the positive one or the negative one?

    Thanks,

    -PFStudent
     
    Last edited by a moderator: May 19, 2008
  2. jcsd
  3. I would justify it by arguing that x is the length of a side of a triangle, which is a non-negative quantity.
     
  4. I'm pretty sure all you have to do is...

    x = a sin(t)
    dx = a cos(t) dt

    I = a S sqrt(1 - sin^2 t) * a cos(t) dt = a^2 S cos^2 t dt

    Now just use a double-angle identity to evaluate the integral and then perform the appropriate substitutions to get back to the solution to the original problem. Not too bad...
     
  5. HallsofIvy

    HallsofIvy 40,930
    Staff Emeritus
    Science Advisor

    Since you are given the integration formula and asked to prove it, you could just differentiate. And, of course, the given formula is defined only for [itex]|x|\le|a|[/itex].
     
  6. Yeah, that's what I was originally thinking too, HallsOfIvy. But does that count as proof? I guess it should. huh.
     
  7. Hey,

    Ok, that makes sense - I kind of thought that reasoning as well. Thanks.

    Thanks,

    -PFStudent
     
Know someone interested in this topic? Share this thead via email, Google+, Twitter, or Facebook

Have something to add?