# 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