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Math Help - help with an integral

  1. #1
    Junior Member
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    help with an integral

    I'm having a lot of trouble with this integration. I'm supposed to simplify the integral so that it looks like one of the equations on a table of integrals, which then can be used to find the integral.

    Integral of (x^3)*sqrt[(4x^2) -(x^4)]

    Thanks.
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  2. #2
    Math Engineering Student
    Krizalid's Avatar
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    Put u=x^2, but that won't solve the problem, you need to manage of the other integrals.
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  3. #3
    Eater of Worlds
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    \int x^{3}\sqrt{4x^{2}-x^{4}}dx

    Rewrite as:

    \int x^{4}\cdot\sqrt{4-x^{2}}dx

    Now, you can use various methods, but trig sub is an option.

    You can look at the tables or try to do it yourself.

    Make the sub x=2sin(t).\;\ dx=2cos(t)dt

    We get 64\int sin^{4}(t)dt-64\int sin^{6}(t)dt

    64\int sin^{4}(t)dt=8\int cos(4t)dt-32\int cos(2t)dt+24\int dt

    Also \int sin^{6}(t)dt=\int\frac{5}{16}dt-\frac{15}{32}\int cos(2t)dt+\frac{3}{16}\int cos(4t)dt-\frac{1}{32}\int cos(6t)dt

    See?. You can break it all up into easier integrals and go from there.

    Of course, as I said, this is one way.
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