Partial fraction integration is a method used in calculus to break down a complex rational function into simpler fractions that can be integrated separately. It is often used to solve integrals involving polynomials.
Partial fraction integration is typically used when the integrand (the function being integrated) is a rational function, meaning it is a ratio of two polynomials. It can also be used to solve improper integrals or when other integration techniques, such as substitution, are not applicable.
To perform partial fraction integration, you first need to factor the denominator of the rational function into linear and irreducible quadratic factors. Then, you set up a system of equations using the coefficients of the factors and solve for the unknown constants. Finally, you integrate each fraction separately and combine the results to get the final answer.
There are two main types of partial fractions: proper fractions and improper fractions. Proper fractions have a degree of the numerator that is less than the degree of the denominator, while improper fractions have a degree of the numerator that is equal to or greater than the degree of the denominator.
Some common mistakes to avoid when using partial fraction integration include not factoring the denominator correctly, making errors in setting up the system of equations, and not checking for extraneous solutions. It is also important to check your final answer by differentiating it to ensure that it is equivalent to the original integrand.