Example 4 (Continued)

We are given

The denominator x² + x + 1 has no real roots because the quadratic formula gives

We therefore proceed immediately to Step 3.

How to do Step 3: The rational function has been broken up into a sum of a polynomial and partial fractions of the two types

(1)

(2) where ax² + bx + c is irreducible.

Polynomials and fractions of type (1) are easily integrated using the Power Rule,

and the rule,

Partial fractions of type (2) can be integrated as follows.

First divide the denominator by an so the fraction has the simpler form

When we make the substitution , we find that

This substitution is called the method of completing the square. Now the integral takes the even simpler form

The first integral can be evaluated by putting w = u² + k², dw = 2u du. The second integral can be evaluated by the trigonometric substitution shown in Figure 8.8.2, u = k tan θ.

Figure 8.8.2