How to Solve the Integral of \(\sqrt[3]{1-x^7}-\sqrt[7]{1-x^3}\)?

[blockcolder]

Homework Statement

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

Homework Equations

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]

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 Statement

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

Homework Equations

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