Finding the solution to a tricky integral, where the primitive function is difficult to find

Question

I am stuck at solving this integral, as I can not find the primitive function of the integrand. Could someone pinpoint me in the right direction?

$$ \int_{0}^{1} \left( \int_{\sqrt[3]{x}}^{1} \frac{1}{\sqrt{1 + y^8}} dy \right) dx$$

Answer 1

The domain of integration is $$x\in[0,1]\text{ and }\sqrt[3] x\le y\le 1$$

This is the same as

$$y\in[0,1]\text{ and }0\le x\le y^3$$

So your integral is

$$\int_0^1\left(\int_0^{y^3}\frac{dx}{\sqrt{1 + y^8}}\right)dy$$ $$=\int_0^1\frac{y^3}{\sqrt{1 + y^8}}\,dy$$

Now substitute $u=y^4$.