How do I find the anti derivative of #(2x^2+5)/(x^2+1)#?
Rewrite the integrand.
# = (2x^2+2)/(x^2+1)+3/(x^2+1)#
# = 2 + 3(1/(x^2+1))#
So the antiderivative is
If you haven't memorized
After substitution, the integral becomes
# = int 1 d theta = theta +C = tan^-1x +C#
If you don't notice how to split the numerator then do the division.
When finding the antiderivative (integral) of a rational function, if the degree of the numerator is greater than the degree of its denominator, division is often a good idea.