A rational function is by definition the quotient of two polynomials. For example

are all rational functions. Remember in the definition of a rational
function, you will not see neither or |*x*| for
example. Note that integration by parts will not be enough to help
integrate a rational function. Therefore, a new technique is needed
to do the job. This technique is called
**Decomposition of
rational functions into a sum of partial fractions** (in short **
Partial Fraction Decomposition**).

Let us summarize the practical steps how to integrate the rational
function :

**1**- If , perform polynomial long-division. Otherwise go to step 2.
**2**- Factor the denominator
*Q*(*x*) into irreducible polynomials: linear and irreducible quadratic polynomials. **3**- Find the partial fraction decomposition.
**4**- Integrate the result of step 3.

** Remark:** The main difficulty encountered in
general when using this technique is in dealing with step 2 and
step 3. Therefore, it is highly recommended to do a serious review of
partial decomposition technique before adventuring into integrating
fractional functions.

The following examples illustrate cases in
which you will be required to use Partial Fraction Decomposition technique:

**
**

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