feedback on my solution (improper integral) i have done this improper integral but i am not sure if i have followed the correct procedure or my answer is correct. Please help!
 

Your answer appears to be correct, though how you got there is unclear.  Where you split the integral in two is technically correct, though it is unclear what purpose you felt that may have served.  From then on, it's fine until you try to plug in, at which point you should have
$$5\times0^{1/5}-5\times(-1)^{1/5}+5\times32^{1/5}-5\times0^{1/5}$$
The first and last terms are both zero, though they would have both cancelled out regardless of where you split the integral.  The remaining terms are equal to $5+10=15$.  However, what you have written down is equal to $-5+10=5$.  Again, it's unclear how you dropped the minus sign and still obtained positive $5$.
