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

Aug 22, 2015

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

#### Explanation:

We can rewrite the fraction

$\frac{x - 1}{1 + {x}^{2}} = \frac{x}{1 + {x}^{2}} - \frac{1}{1 + {x}^{2}}$

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.