You're not doing the integration by parts correctly. Your derivative of u is incorrect (you need the chain rule), and your integral of dv is incorrect (use a substitution).
Isn't d/dx of e^5x = e^4x * 5 ?
Also, I list it as simply du = e^5x dx because my teacher does that for some reason. For instance in the example of e^x he lists it as e^x = e^x dx. So I did the same.. I don't want to learn something the wrong way here I thought my way of showing it was correct.
So if I were to fix this would the rest be ok?