Difficult integral (for me)

Homework Statement

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

None

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?

Simon Bridge
Homework Helper
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?

Ray Vickson
Homework Helper
Dearly Missed

Homework Statement

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

None

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?

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

RGV