wow, very neat writing.
the result is correct, but I think you can save some steps so that you can save some time. especially when you r having an exam.
I think you can do more excercise to get use to this type of question.
You should skip the entire integration by substitution. You know that it will be something to do with ln(4+x^2) so write that and then work out what constants you need to add to make the integral work.