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Difficult integral (for me)

  1. Nov 19, 2011 #1
    1. The problem statement, all variables and given/known data

    [itex]\int_0^1 \sqrt[3]{1-x^7}-\sqrt[7]{1-x^3} dx[/itex]

    2. Relevant equations

    None

    3. The attempt at a solution

    I tried using the substitutions [itex]u=\sqrt[3]{1-x^7}[/itex] and [itex]u=\sqrt[7]{1-x^3}[/itex] to no avail and I couldn't think of any more substitutions. Any suggestions?
     
  2. jcsd
  3. Nov 19, 2011 #2

    Simon Bridge

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    express the roots as fractional powers
    treat the two terms separately

    consider the geometry of the integral in terms of the relation
    [itex]y^3 + x^7 = 1[/itex]

    the roles of x and y swap in the second integral don't they?
     
  4. Nov 19, 2011 #3

    Ray Vickson

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    In the first integral, let y = x^7. Then look up "Beta function".

    RGV
     
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