Partial Fraction Decomposition

Partial Fraction Decomposition is a method used to express a rational function as the sum of simpler fractions, making it easier to integrate or analyze.

The first step is to ensure that the degree of the numerator is less than the degree of the denominator. If not, perform polynomial long division.

Set up the partial fractions as $$\frac{A}{(x-a)} + \frac{B}{(x-b)} + ...$$ for each linear factor $$(x-a)$$ , where $$A, B, ...$$ are constants to be determined.

For repeated linear factors like $$(x-a)^n$$ , include terms like $$\frac{A_1}{(x-a)} + \frac{A_2}{(x-a)^2} + ... + \frac{A_n}{(x-a)^n}$$ .

Multiply both sides by the common denominator, then equate coefficients or substitute convenient values for $$x$$ to solve for the constants.

For irreducible quadratic factors like $$ax^2 + bx + c$$ , use the form $$\frac{Ax + B}{ax^2 + bx + c}$$ .

It simplifies the integration of rational functions by breaking them down into simpler fractions that are easier to integrate.

$$\frac{2x+3}{(x-1)(x+2)} = \frac{A}{x-1} + \frac{B}{x+2}$$ , where you solve for $$A$$ and $$B$$ .

The decomposition is $$\frac{6}{x-4} - \frac{7}{x+1}$$ .

The degree of the numerator must be less than the degree of the denominator to apply Partial Fraction Decomposition directly.