Your last line should actually look like:
now use: .
Then rationalize the denominator and simplify.
We also have the identity that
Now we need to integrate
FYI most people just memorize this anti-derivative but here is the substitution to work it out. We will do it in two steps. first let
Now here is the part most folks memorize.
This gives that
now the integral becomes
Argh! I know I'm tremendously slow, but now that I see it, it's rather trvial.
Thinking of it my rationalization was different and wrong. I used the incorrect conjugate.
Ah, is TheEmpySet's method another approach or was he just hinting at what is directly above this post?
Thanks for your patience.