To find we can simplysubstituteany valueof alongwith the valuesof and obtained.
Choose :
Now we havesolvedfor and We use the partialfractiondecompositionto integrate.
.
Example4:
This problemis an exampleof an improperrationalfunction.Evaluatethe definiteintegral
.
Solution:
This rationalfunctionis improperbecauseits numeratorhas a degreethat is higherthan its denominator.
The first step is to dividethe denominatorinto the numeratorby long divisionand obtain
Now applypartialfunctiondecompositiononly on the remainder,
As we did in the previousexamples,multiplyboth sidesby and then set and to
obtainthe basicequation
For