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Math Help - Difficult integrals

  1. #1
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    Difficult integrals

    \int\frac{xsinx}{1+(cosx)^{2}}dx
    \int_{-\infty }^{\infty }e^{-x^{2}}dx
    \int\frac{\sqrt{\sqrt{x^4 + 1}-x^{2}}}{x^4 + 1}dx
    \int\frac{\sqrt{x}ln(x)e^{x}x^{3}}{\sqrt{lnx}}dx
    I have been working on these integral for 4 weeks... haven't been able to get anything yet. I am preparing for a competition and came across these bad boys.
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    Re: Difficult integrals

    I found the solution to the second integral. It turned out to be the Gaussian Integral!
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  3. #3
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    Re: Difficult integrals

    Try using this page. Wolfram Mathematica Online Integrator
    I ve done for you the first one. Doesnt seem to have any sense but maybe I made a typing mistake.

    http://integrals.wolfram.com/index.jsp?expr=(x*sin(x))%2F(1+%2B+cos(x)^2)&rando m=false
    Last edited by kezman; March 21st 2013 at 01:11 PM.
    Thanks from smokesalot
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  4. #4
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    Re: Difficult integrals

    Quote Originally Posted by smokesalot View Post
    \int\frac{xsinx}{1+(cosx)^{2}}dx
    \int_{-\infty }^{\infty }e^{-x^{2}}dx
    \int\frac{\sqrt{\sqrt{x^4 + 1}-x^{2}}}{x^4 + 1}dx
    \int\frac{\sqrt{x}ln(x)e^{x}x^{3}}{\sqrt{lnx}}dx
    I have been working on these integral for 4 weeks... haven't been able to get anything yet. I am preparing for a competition and came across these bad boys.
    \displaystyle \int{\frac{x\sin{(x)}}{1 + \cos^2{(x)}}\,dx}

    Use integration by parts with \displaystyle u = x \implies du = dx and  dv = \frac{\sin{(x)}}{1 + \cos^2{(x)}} \implies v = -\arctan{ \left[ \cos{(x)} \right] } and the integral becomes

    \displaystyle \int{\frac{x\sin{(x)}}{1 + \cos^2{(x)}}\,dx} = -x\arctan{\left[ \cos{(x)} \right]} + \int{\arctan{ \left[ \cos{(x)} \right]} \,dx}

    This does not have a closed-form solution in terms of elementary functions.
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    Re: Difficult integrals

    wow nice, didn't see that! Thank you so much
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