How would i go about using partial fraction decomposition on the integral, (int) (x+2)/(x^2+4x+5) dx.

Is there a way to do this utilizing complex number decomposition.

3 answers

You don't want to dive into this with complex numbers. Partial fractions will not help here.

Let
u = x^2+4x+5
du = 2(x+2) dx
and that is
∫ 1/2 du/u = 1/2 lnu = 1/2 ln(x^2+4x+5) + C
Unfortunately I have to make sure that the normal computation and the complex computation match up
well, you can go ahead and do that, but you better make sure you understand something about the inverse hyperbolic functions, especially argtanh(x), which is defined with complex logs.