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Math Help - Another fun integral

  1. #1
    MHF Contributor Mathstud28's Avatar
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    Another fun integral

    \int_0^{\infty}\bigg[\frac{\sin^2(px)\cos^2(px)}{x^2}-\frac{\sin^4(px)}{3x^2}\bigg]~dx
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by Mathstud28 View Post
    \int_0^{\infty}\bigg[\frac{\sin^2(px)\cos^2(px)}{x^2}-\frac{\sin^4(px)}{3x^2}\bigg]~dx
    I got to this point...and then I'm kinda stuck...any hints?

    \int_0^{\infty}\frac{\sin^2(px)}{x^2}\,dx-\frac{4}{3}\int_0^{\infty}\frac{\sin^4(px)}{x^2}\,  dx

    --Chris
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  3. #3
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by Chris L T521 View Post
    I got to this point...and then I'm kinda stuck...any hints?

    \int_0^{\infty}\frac{\sin^2(px)}{x^2}\,dx-\frac{4}{3}\int_0^{\infty}\frac{\sin^4(px)}{x^2}\,  dx

    --Chris
    Going that route?

    \frac{1}{x^2}=\int_0^{\infty}ye^{-yx}~dy
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  4. #4
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by Chris L T521 View Post
    I got to this point...and then I'm kinda stuck...any hints?

    \int_0^{\infty}\frac{\sin^2(px)}{x^2}\,dx-\frac{4}{3}\int_0^{\infty}\frac{\sin^4(px)}{x^2}\,  dx

    --Chris
    I didn't just edit my quote so you would see this, I went through the calculations of this and I think it is wrong
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  5. #5
    MHF Contributor Mathstud28's Avatar
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    Hint:

    Let \int_0^{\infty}\bigg[\frac{\sin^2(px)\cos^2(px)}{x^2}-\frac{\sin^4(px)}{3x^2}\bigg]~dx=f(p)
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  6. #6
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by Mathstud28 View Post
    \int_0^{\infty}\bigg[\frac{\sin^2(px)\cos^2(px)}{x^2}-\frac{\sin^4(px)}{3x^2}\bigg]~dx
    No one's going to bite?

    This is

    \frac{1}{12}\frac{d^2}{dp^2}\int_0^{\infty}\frac{\  sin^4(px)}{x^4}~dx=\frac{\pi{p}}{6}
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