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Math Help - need help with integration

  1. #1
    Newbie
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    Feb 2009
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    need help with integration

    Int (3x^3.5)/(3+x^2.5) dx

    I tried using integration by parts and the equation uv-int (vdu) and im not getting the right answer. Could someone please show me the steps on how to do this question?
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  2. #2
    Senior Member
    Joined
    Dec 2007
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    Anchorage, AK
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    Have you considered making the u-substitution u=\sqrt{x}? Then dx=2u\,du, and your integral becomes
    \int\frac{3x^{7/2}}{3+x^{5/2}}\,dx=\int\frac{6u^8}{3+u^5}\,du
    =\int{6u^3}-\frac{18u^3}{3+u^5}\,du

    The first term is easily integrated, leaving the rational function \frac{18u^3}{3+u^5} to be integrated.
    Using the fifth roots of unity, one can factor the denominator above into real linear and quadratic factors:
    (u^5+3)=(u+3^{1/5})(u^2-3^{1/5}\frac{1-\sqrt5}{2}u+3^{2/5})(u^2-3^{1/5}\frac{1+\sqrt5}{2}u+3^{2/5})

    One can then use partial fractions to separate the above into rational functions with linear or quadratic denominators, which you should be able to integrate.

    --Kevin C.
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