I've solved this problem, but it's more of a nit-pick on something I tried previously in that according to the book was not correct, and now I'm just in need of someone's expertise on why I was not allowed to do this.
Anyway, the integral and solution is: -
In this case, I tried a substitution to begin with: -
Now, on the second attempt I made the appropriate double-angle substitution and achieved the solution I should have done. But, I can't see what is wrong with the former way I chose instead? Why can't I do it this way?