Note that when you are solving a definite integral by making a substitution, you do not have to back-substitute for u to get an integral in terms of x. It is much better for many reasons (such as less time, less chance of mistake etc.) to stay with the new variable.
So you should be substituting your (correct) integral terminals into the u-integral that you calculate rather than trying to solve it in terms of x and the original terminals. Otherwise, why even bother getting new integral terminals if you're never going to use them. But, if you're determined to do it this way, then the anti-derivative you state is correct, so why are you even worrying about the u-integral terminals. Just use the x-integral terminals in your answer.
And did you consider the alternative substitution I suggested at all?
Ah, I did consider the substitution you recommended. However, the section I am working on is trigonometric substitution. As such, I feel it would be best to stick with the trig substitution here.
So, if I am understanding this correctly... if I get my anti-derivative in terms of x, then there is no need to calculate new terminals? In this case my answer would simply be ?
What is the purpose of finding new terminals in this case?
I got my new terminals as shown below:
Ah... does this look a little more correct?
I just caught my miscalculation upon rereading my post.
My book shows the act of finding new terminals as an important step, yet displays all "final" anti-derivatives in terms of x. Is there a reason that they would do this?
Should finding my answer with respect to theta be easier than solving in terms of x?
When I look at the following:
from to the answer is not obvious to me. Is there another rule or identity that I should be using?
I am really struggling to understand all of this again, and I do appreciate your patience with my questions.