I made the substitution . I then differentiated v with respect to u in order to get an integral in terms of dv.
The limits changed because I was integrating with respect to v. There is no reason why I could have left the original limits and then substituted u back in but I find changing the limits easier.
I found the limits by changing the existing limits using the expression for v.
Skeeter's method works just as well and is probably simpler for this problem