Compute the following by reversing the order of integration:

$\displaystyle \int_0^1\int_1^x xye^{y^4} dy dx$

Reversing the order, that's:

$\displaystyle \int_0^1\int_y^1 xye^{y^4} dx dy$

I got to a stage where I have:

$\displaystyle \frac{1}{2}\int_0^1 e^{y^4}(y-y^3) dy$.

Is that correct so far?

I don't know if I can integrate this any further though... Any idea?