Homework Help: Difficult integral (for me)

1. Nov 19, 2011

blockcolder

1. The problem statement, all variables and given/known data

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

2. Relevant equations

None

3. The attempt at a solution

I tried using the substitutions $u=\sqrt[3]{1-x^7}$ and $u=\sqrt[7]{1-x^3}$ to no avail and I couldn't think of any more substitutions. Any suggestions?

2. Nov 19, 2011

Simon Bridge

express the roots as fractional powers
treat the two terms separately

consider the geometry of the integral in terms of the relation
$y^3 + x^7 = 1$

the roles of x and y swap in the second integral don't they?

3. Nov 19, 2011

Ray Vickson

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

RGV