$\displaystyle
\int_0^1 x^2e^{x^{1/5}}~dx
$
I assume that this is solved using integration by parts, but what should I choose for u and dv?
Yep you would just solve the problem like normal and you don't have to worry bout swapping $\displaystyle x^{\frac{1}{5}}$ for w because all you have to do is change your lower and upper bound to correlate with w.
For example
Your lower bound
$\displaystyle 0^{\frac{1}{5}} = 0$
and upper bound
$\displaystyle 1^{\frac{1}{5}} = 1$
$\displaystyle 5 \int_0^1 w^{14}e^w~dw$
$\displaystyle \int_0^1 x^2 e^{x^{1/5}}~dx ~=~ 5\int_0^1 u^{14} e^u~du$
$\displaystyle \int u^{14}e^u~du = u^{14} e^u - 14 u^{13}e^u + 14.13 u^{12}e^u ..... - 14! u^1 e^u+14! e^u~+$ $\displaystyle C = \sum_{i=0}^{14}u^i e^u (-1)^i \frac{14!}{i!}~+C$
Evaluating this from 0 to 1 and multiplying by 5 gives us $\displaystyle 5\left [ \left ( \sum_{i=0}^{14} e (-1)^i \frac{14!}{i!} \right )-14! \right ] \approx 0.8526185$