Final answer:
To find the partial fraction decomposition of a rational function, factor the denominator into irreducible factors and express the rational function as the sum of its partial fractions.
Step-by-step explanation:
A system of equations can be used to find the partial fraction decomposition. Let's say we have a rational function in the form of p(x)/q(x), where p(x) and q(x) are polynomials. To find the partial fraction decomposition, we need to factor the denominator q(x) into its irreducible factors. Then we express the rational function as the sum of its partial fractions, where each partial fraction has a denominator corresponding to one of the irreducible factors.
For example, if we have the rational function 2x^2 + 5x + 3 / (x^2 - 2x - 3), we can factor the denominator as (x - 3)(x + 1). So the partial fraction decomposition would be A/(x - 3) + B/(x + 1), where A and B are constants that we can solve for by finding a common denominator and equating the numerators.
Once we have the partial fraction decomposition, we can integrate each of the partial fractions individually to find the integral of the original rational function.