How do you use partial fraction decomposition to decompose the fraction to integrate $\frac{x - 1}{1 + x^{2}}$ $(x-1)/(1+x^2)$ ?

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Answer 1 · 2015-08-22T16:44:02

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

We can rewrite the fraction

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

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

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

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