I realize this is a challenging example. If there are any questions, please leave a comment, I will answer them.
I solved this problem using a given formula for the integration of exponential functions.
Although this method works for most all exponential integration problems, I will still upload another post showing how you would do this problem using a 'u' substitution method.
This method is a lot faster than 'u' substitution, and it would benefit you to go for it in an exam type situation.