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Integrate sin(a*x^2) from -a to a

  1. May 22, 2008 #1
    Hey all, this should be a simple integral, but I cant seem to solve it. I tried some simple u-subs and integration by parts, but to no avail. Also, maple throws out some Fresnel function.

    [tex]\int_{-a}^{a} x^{2} cos\left( a x^{2} \right) dx[/tex]
     
  2. jcsd
  3. May 22, 2008 #2

    Defennder

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    Well, that should tell you it doesn't have an anti-derivative expressible in terms of elementary functions.
     
  4. May 22, 2008 #3
    I was hoping that perhaps it had a nice representation due to the limits of integration.

    edit: I also don't particularly trust maple, as there have been times in the past where it couldn't do integrals that I could. (or perhaps I am using it wrong)
     
    Last edited: May 22, 2008
  5. May 22, 2008 #4
    From the looks of it, it's symmetric, so you can change the limits of integration from 0 to a and multiplying your integral by 2. Idk if that helps at all or not :p
     
  6. May 22, 2008 #5
    This is not a homework question. I am not a student. Please move the question back to the appropriate forum.
     
  7. May 22, 2008 #6

    Defennder

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    Well if it were an odd function, we would be able to conclude that the answer is 0 without having to work out the anti-derivative. But this is an even function.

    By the way, if you want your thread to be moved back you should request a PF mentor by private messaging to do so.
     
  8. May 22, 2008 #7
    >> syms x;
    >> f = x^2*cos(x^2);
    >> int (f,x)

    ans =

    1/2*sin(x^2)*x-1/4*2^(1/2)*pi^(1/2)*FresnelS(2^(1/2)/pi^(1/2)*x)


    >> int(f,x,-1,1)

    ans =

    sin(1)-1/2*2^(1/2)*pi^(1/2)*FresnelS(2^(1/2)/pi^(1/2))

    Matlab did same ><
    I wanted to prove that Matlab is better

    "edit: I also don't particularly trust maple, as there have been times in the past where it couldn't do integrals that I could. (or perhaps I am using it wrong)"
    sometimes, you need to put the function in integrable form ... (computer's dun think)
     
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