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Home Exponential and Logartihmic Functions Integration of Rational Functions Examples Example 4
 
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Example 4

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Step 1

No long division is needed.

Step 2

The denominator x2 + x + 1 cannot be factored, i.e., it is irreducible. In this case no sum of partial fractions is needed.

Step 3

To integrate 08_exp-log_functions-451.gif

we use the method of completing the square. We have x2 + x + 1 = (x + ½)2 + ¾. Let u = x + ½. Then du = dx and

08_exp-log_functions-452.gif

We used the trigonometric substitution illustrated in Figure 8.8.1.

08_exp-log_functions-453.gif

08_exp-log_functions-454.gif

Figure 8.8.1

How to do Step 2: We wish to break a quotient R(x)/G(x) into a sum of partial fractions. First, factor the denominator G(x) into a product of linear terms of the form ax + b, and irreducible quadratic terms of the form ax2 + bx + c. It can be proved that every polynomial can be so factored, but we shall not give the proof here. Two theorems from elementary algebra are useful for factoring a given polynomial.
Home Exponential and Logartihmic Functions Integration of Rational Functions Examples Example 4

Last Update: 2006-11-15