Integrals of Rational Functions

Simple Proper Rational Functions

Proper rational functions are functions whose degree of numerator is less then that of its denominator, for example:

$y = \frac{1}{(x + 1)^{2}}, y = \frac{x + 1}{x^{2} + 3 x + 8}$

To find integrals of simple proper rational functions, substitution can help us handle with most cases.

For those improper rational functions, simplify them into a polynomial with a proper rational function.

When facing items $(a x + b)^{k}$ in the denominator, decompose it into $\frac{A_{1}}{(a x + b)} + \frac{A_{2}}{(a x + b)^{2}} + \dots + \frac{A_{k}}{(a x + b)^{k}}$ , where $A_{1}, A_{2}, \dots A_{k}$ are constants that can be determined.
When facing items $(a x^{2} + b x + c)^{k}$ in the denominator, decompose it into $\frac{B x_{1} + C_{1}}{a x^{2} + b x + c} + \frac{B x_{2} + C_{2}}{(a x^{2} + b x + c)^{2}} + \dots + \frac{B x_{k} + C_{k}}{(a x^{2} + b x + c)^{k}}$ .

Then most complex rational functions can be simplify into the sum of many simple proper rational functions that are easy to integrate.