Would you mind revealing in what class you saw this problem? I am just curious which classes at the graduate level test students on their calculus abilities. Also, was this already set up or was it part of a word problem? Any more information on this would be very interesting.
Here is what I did:
The function should have the following form:
A/(x^2+1) + B/(X^2+4)
then let A = 1 => solve for B:
=> B = -4/(x^2+1)
substitute B back into the function
=> we have something like:
1/(x^2+1) - 4/[(x^2+4)*(x^2+1)]
The second integral need to split up again . My question is how do you know a proper way to decompose it (as you did)?