Question

How do you use partial fraction decomposition to decompose the fraction to integrate `(x-1)/(1+x^2)`?

Answer 1

Because the denominator is already irreducible over the Reals, the fraction cannot be further decomposed. But . . .

Explanation:

We can rewrite the fraction

`(x-1)/(1+x^2) = x/(1+x^2) - 1/(1+x^2)`

The first fraction can be integrated by substitution: `u = 1+x^2`

The second needs either a trig substitution or recognizing the derivative of arctan.