Well, I can refer you to the answer:
http://integrals.wolfram.com/index.jsp?expr=e^x^%281%2F3%29&random=false
complete the integral e^(x^1/3)
4 answers
here, click on "show steps"
http://www.wolframalpha.com/input/?i=integrate+e^x^%281%2F3%29
http://www.wolframalpha.com/input/?i=integrate+e^x^%281%2F3%29
in other words substitute u = x^(1/3)
du = (1/3) x^-(2/3) dx
then integrate by parts
du = (1/3) x^-(2/3) dx
then integrate by parts
thanks