Improper Rational Functions and Long Division

If the numerator $P(x)$ has degree greater than or equal to the degree of the denominator $Q(x)$, then the rational function $\displaystyle\frac{P(x)}{Q(x)}$ is called improper. In this case, we use long division of polynomials to write the ratio as a polynomial with a remainder.

If dividing $P(x)$ by $Q(x)$ gives $S(x)$ with remainder $R(x)$, then the degree of the $R(x)$ is less than the degree of $Q(x)$ as a result of the long division.  We have

Integrating $S(x)$ is easy, since it's a polynomial, and we can use partial fractions on the proper rational function $\displaystyle\frac{R(x)}{Q(x)}$.

This long division of polynomials and the subsequent partial fraction decomposition and integration is explained in the following video.