Mastering the technique of partial fractions is a cornerstone when integrating complex rational functions. This method simplifies these functions into a sum of simpler fractions that are more straightforward to integrate.
Decomposition into Partial Fractions
To integrate a complex rational function:
1. Break it down into simpler fractions with linear or quadratic denominators using partial fractions.
2. Integrate each simpler fraction separately.
Example:
For :
Express as partial fractions:
Integration After Decomposition
Integrate a function by partial fractions:
1. Decompose into simpler fractions.
2. Integrate each fraction.
Example:
For :
Decomposed form:
Integrated result:
With limits applied: