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Math Help - Help evaluating integral

  1. #1
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    Help evaluating integral

    Could someone please help evaluate the integral \int {x}^{5}\sqrt{1-{x}^{3} }
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  2. #2
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    Quote Originally Posted by dnftp View Post
    Could someone please help evaluate the integral \int {x}^{5}\sqrt{1-{x}^{3} }
    Let u = 1 - x^3

    du = - 3x^2 dx

    So we have,
    -\frac{1}{3}\int {x}^{3}\sqrt{u} \ du

    But we said u = 1 - x^3, so rearranging we get x^3 = (1 - u)

    So sub that in:

    = -\frac{1}{3}\int(1 - u)\sqrt{u} \ du

    = -\frac{1}{3}\int\sqrt{u} - u^{3/2} \ du

    Now it's easy. Just remember to sub in the value of u after you've done the integration.
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  3. #3
    MHF Contributor FernandoRevilla's Avatar
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    Alternatively, we can consider it as a binomial differential


    I = \int x^m(a+b x^n)^p\;dx\quad (m,n,p\in\mathbb{Q})

    In our case

    \dfrac{m+1}{n}\in\mathbb{Z}


    So, with the substitution

    a+bx^n=t^s\;,\quad s=\textrm{denom}(p)

    we obtain a rational integral on t.
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