Methods of Integration: Integration Using Partial Fractions

A rational function is defined as the ratio of two polynomials in the form ${\displaystyle \frac{P\left(x\right)}{Q\left(x\right)}}$ , where P (x) and Q(x) are polynomials in x and Q(x) ≠ 0. If the degree of P(x)  is less than the degree of Q(x), then the rational function is called proper, otherwise, it is called improper. The improper rational functions can be reduced to the proper rational  functions by long division process. Thus, if  ${\displaystyle \frac{P\left(x\right)}{Q\left(x\right)}}$ is improper ,then ${\displaystyle \frac{P\left(x\right)}{Q\left(x\right)}}$ = T(x) + ${\displaystyle \frac{P_1\left(x\right)}{Q\left(x\right)}}$
where T(x) is a polynomial in x and ${\displaystyle \frac{P_1\left(x\right)}{Q\left(x\right)}}$ is a proper rational function. As we know how to integrate polynomials, the integration of any rational function is reduced to the integration of a proper rational function. The rational functions which we shall consider here for integration purposes will be those whose denominators can be factorised into linear and quadratic factors. Assume that we want to evaluate ${\displaystyle \int \frac{P\left(x\right)}{Q\left(x\right)}}$ dx , where ${\displaystyle \frac{P\left(x\right)}{Q\left(x\right)}}$ is proper rational function.  It is always possible to write the integrand as a sum of simpler rational functions by a method called partial fraction decomposition.  The above table shows that the types of simpler partial fractions that are to be associated with various kind of rational functions.

Point (5) ${\displaystyle x^2+bx+c}$ cannot be factorised further