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Calculate the antiderivatives

  1. Nov 3, 2012 #1
    1. The problem statement, all variables and given/known data
    [itex]\int \frac{ln(x^{2}+4^{x})}{\sqrt{3x^{7}+7x^{^3}}}dx[/itex]


    2. Relevant equations
    X.


    3. The attempt at a solution
    Wolfram Alpha seem to give no answer.
     
    Last edited by a moderator: Jun 19, 2014
  2. jcsd
  3. Nov 3, 2012 #2

    tiny-tim

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    the x2 + 4x makes that impossible to do by analytic means
     
  4. Nov 3, 2012 #3
    Re: [itex]\int \frac{ln(x^{2}+4^{x})}{\sqrt{3x^{7}+7x^{^3}}}dx[/itex]

    Actually my original problem was determining the convergence or divergence of the following improper integral:
    [itex]\int^{+∞}_{0} \frac{ln(x^{2}+4^{x})}{\sqrt{3x^{7}+7x^{^3}}}dx[/itex]
    I split the integral into
    [itex]\int^{+∞}_{1} \frac{ln(x^{2}+4^{x})}{\sqrt{3x^{7}+7x^{^3}}}dx[/itex]
    and [itex]\int^{1}_{0} \frac{ln(x^{2}+4^{x})}{\sqrt{3x^{7}+7x^{^3}}}dx[/itex]
    , calculate the antiderivatives, then evaluate the limit of them.
    So if I can't calculate the antiderivative, is there any alternative way to see if this integral convergent or not ??
     
  5. Nov 3, 2012 #4
    Re: [itex]\int \frac{ln(x^{2}+4^{x})}{\sqrt{3x^{7}+7x^{^3}}}dx[/itex]

    Tiny-Tim, how did you know thats impossible to do by analytical means? What should I google to learn more?
     
  6. Nov 3, 2012 #5
    Do you know how to find the asymptotic behavior of your integrand for [itex]x \rightarrow \infty[/itex] and [itex]x \rightarrow 0[/itex]? If yes, then you may use the comparison test.
     
  7. Nov 4, 2012 #6

    tiny-tim

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    hi superg33k! :smile:
    it's obvious just from looking at it … that bracket is simply too complicated for any of the known techniques to work! :redface:

    once you've had lots of practice at differentiating and integrating, you'll see why :smile:
     
  8. Nov 4, 2012 #7
    I'll start off. Let us consider the upper bound first. For [itex]x \rightarrow \infty[/itex]. [itex]x^2 = o(4^x)[/itex], so the logarithm in the numerator behaves as [itex]\sim x \, \ln(4)[/itex]. Similarly, [itex]7 x^3 = o(3 x^7)[/itex], so the expression under the square root in the denominator behaves as [itex]\sim 3 x^7[/itex]. Therefore, the integrand behaves as:
    [tex]
    \sim \frac{x \, \ln(4)}{\sqrt{3 x^7}} = \frac{\ln(4)}{\sqrt{3}} \, x^{-5/2}
    [/tex]
    Do you know whether the integral:
    [tex]
    \int_{1}^{\infty}{x^{-5/2} \, dx}
    [/tex]
    is convergent or divergent?

    A similar analysis can be done on the lower bound of the integral. However, what are the dominant terms in this limit?
     
  9. Nov 4, 2012 #8
    Re: [itex]\int \frac{ln(x^{2}+4^{x})}{\sqrt{3x^{7}+7x^{^3}}}dx[/itex]

    Thank you very much, I can take it from here :D.
     
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