Why do I have to set up the partial fractions like this?

[Lo.Lee.Ta.]

1. ∫[(x⁴ + x + 1)/(x(x² + 1))]dx


2. When I first did this problem, I divided and got:
∫[x + (-x² + x + 1)/(x³ + x)]dx

(x³ + x) = x(x² + 1)

I then set up the fraction as: A/x + B/(x² + 1)

BUT, the solution to this problem says: A/x + [(Bx + C)/(x² + 1)]

How would I know to use Bx + C? Where did the x come from and why do we need the C?

Please let me know what situations need this new format.
Thank you so much! :D

 

[rock.freak667]

For every quadratic factor like ax^2+bx+c which cannot be expressed in terms of real solutions, the partial fractions is

(Ax+B)/(ax^2+bx+c)

 

[Dick]

And the reason you need it is because A/x + B/(x^2 + 1) there is no choice of A and B that will make that equal to (-x^2 + x + 1)/(x^3 + x). You really need the extra parameter.