Integration of Rational Fractions | Techniques of Integration

Partial Fractions

Functions of $x$ that can be expressed in the form $P(x) / Q(x)$, where both $P(x)$ and $Q(x)$ are polynomials of $x$, is known as rational fraction. A rational fraction is known to be a proper fraction if $P(x)$ is less in degree power than $Q(x)$. Example of proper fraction is...

$\dfrac{2x^2 + 4x - 5}{5x^3 + 6x^2 -2x -1}$


A rational fraction is said to be an improper fraction if the degree of $P(x)$ is greater than or equal to the degree of $Q(x)$. Examples are...

$\dfrac{3x^2 - 2x + 1}{2x^2 + 6}$ and $\dfrac{4x^2 - 2x + 3}{3x + 2}$


Improper fraction may be expressed as the sum of a polynomial and a proper fraction. For example:

$\dfrac{12x^2 - 13x - 9}{4x - 7} = 3x + 2 + \dfrac{5}{4x - 7}$


Proper fraction such as $(x - 4) / (2x^2 - 4x)$ can be expressed as the sum of partial fractions, provided that the denominator will factorized.

Integration of any rational fraction depends essentially on the integration of a proper fraction by expressing it into a sum of partial fractions. There are four cases that may arise in dealing with integrand involving proper fraction.