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Math Help - Interesting Integral

  1. #1
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    Interesting Integral

    Compute
    \int_0^{\pi/2}\frac{\sin x}{9+16\sin(2x)}\,dx
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  2. #2
    MHF Contributor chisigma's Avatar
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    Re: Interesting Integral

    Use the substitutions...

    t= \tan \frac{x}{2}

    dx = 2\ \frac{dt}{1+t^{2}}

    \sin x = \frac{2 t}{1+t^{2}}

    \cos x = \frac{1-t^{2}}{1+t^{2}}

    ... and take into account the identity...

    \sin 2x = 2\ \sin x\ \cos x

    Kind regards

    \chi \sigma
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  3. #3
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    Re: Interesting Integral

    I am not sure with that. The rational function will be very ugly ...
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    Forum Admin topsquark's Avatar
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    Re: Interesting Integral

    Quote Originally Posted by watchmath View Post
    I am not sure with that. The rational function will be very ugly ...
    Not so bad, though, from a Calculus standpoint. Notice that the denominator is a biquadratic: at^4 + bt^2 + c. Use a substitution u = t^2 and the form will be much more recognizable.

    -Dan
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  5. #5
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    Re: Interesting Integral

    I got
    \frac{4t}{9t^4-64t^3+18t^2+64t+9}
    and I would say that is ugly .
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  6. #6
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    Re: Interesting Integral

    Quote Originally Posted by watchmath View Post
    I got
    \frac{4t}{9t^4-64t^3+18t^2+64t+9}
    and I would say that is ugly .
    The bottom factorises to \displaystyle (t^2 - 8t + 9)(9t^2 + 8t + 1), so the next step is partial fractions.
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  7. #7
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    Re: Interesting Integral

    Quote Originally Posted by Prove It View Post
    The bottom factorises to \displaystyle (t^2 - 8t + 9)(9t^2 + 8t + 1), so the next step is partial fractions.
    How did you obtain this factorisation, if you don't mind me asking? I'm not seeing it; neither can I find a clever/painless way of solving the integral!
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  8. #8
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    Re: Interesting Integral

    Quote Originally Posted by TheCoffeeMachine View Post
    How did you obtain this factorisation, if you don't mind me asking? I'm not seeing it; neither can I find a clever/painless way of solving the integral!
    I used Wolfram Alpha, lol...
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  9. #9
    Senior Member bugatti79's Avatar
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    Re: Interesting Integral

    Quote Originally Posted by Prove It View Post
    I used Wolfram Alpha, lol...
    lol!!
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