Definite Integral Arising from a Double Integral

Question

I gave an integral to a student. She reported back to me that she could not do it. I've tried a couple of approaches and have failed. I imagine it is fairly easy. It's a double integral. And, no matter, which variable you do first, you end up with an integral like: $$ \int_1^2 \frac{1}{3x}e^{x^5}\,dx. $$ Any help would be appreciated. In case I have messed up the initial integrations, the original integral was: $$ \int_1^2\int_1^3 x^4y^2e^{x^5y^3}\,dy\,dx. $$

Answer 1

This integral pertains to the exponential integral function, $\text{Ei}(z)$, so you weren't going to get it done by hand.

Mathematica confirms:

Just to make sure, let's check the double integral as well:

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PS Note the particular definition Mathematica employs here for $\text{Ei}(z)$: