## Example 7.2.12(c): Integrating Rational Functions

This time it seems that we can not use partial fraction decomposition because the degree of the numerator is higher than that of the denominator. To remedy that problem, we use "long division" to divide the polynomials:Integrating= x + 2 +

*x + 2*is easy, so the problem is reduced to finding

*dx*and the partial fractions decomposition theorem applies just fine to this integrand. We know that

Therefore we get three equations in three unknowns:

Solving this system of equations gives

-3B + 2C = 12 (for the constant coefficient) -3A + B = 2 (for the xcoefficient)A + C = 4 (for the xcoefficient)^{2}

Therefore the one complicated integral above changes into three simpler ones:A = -10/11, B = -8/11, C = 54/11

The second integral is easy (involving the

*ln*). The first integral seems to be somewhat difficult, because

*arctan(x) = 1 / (1 + x*which does not quite work, and the numerator is

^{2})*not*the derivative of the denominator, so the

*ln*is out, too. But a little algebra and some substitution will do the trick:

Taking everything together gives: