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Math Help - Convergence of the integral

  1. #1
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    Convergence of the integral

    I need help please how to solve this exercise if the integral is convergence or not


    1/(x^(1/3) + x^(3/2)) dx
    0


    thanks a lot!
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by tukilala View Post
    I need help please how to solve this exercise if the integral is convergence or not


    1/(x^(1/3) + x^(3/2)) dx
    0


    thanks a lot!
    Note that \frac{1}{\sqrt[3]{x}+\sqrt[2]{x^3}}\leqslant\frac{1}{\sqrt[2]{x^3}}. Now, what can you say about the infinity end of the integral? Similarly \frac{1}{\sqrt[3]{x}+\sqrt[2]{x^3}}\leqslant\frac{1}{\sqrt[3]{x}}, so what can you say about the zero side?
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  3. #3
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     \int_0^{\infty} \frac{dx}{ x^{1/3} + x^{3/2} }

     = \int_0^1 \frac{dx}{ x^{1/3} + x^{3/2} } + \int_1^{\infty} \frac{dx}{ x^{1/3} + x^{3/2} }

     \leq \int_0^1 \frac{dx}{ x^{1/3} } + \int_1^{\infty} \frac{dx}{ x^{3/2}}
    Last edited by simplependulum; June 9th 2010 at 02:54 AM.
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