Welcome to Section 7-4: Integration of Rational Functions by Partial Fractions!

Alright class, let's tackle a new integration technique! In this section, we'll be focusing on integrating rational functions. Remember, a rational function is just a ratio of two polynomials, like this: $f(x) = \frac{P(x)}{Q(x)}$ where $P(x)$ and $Q(x)$ are both polynomials.

The core idea here is to decompose a complex rational function into a sum of simpler fractions that we *already* know how to integrate. It's like taking a complicated dish and breaking it down into its individual ingredients. Think of it like this:

$$ \frac{x+5}{x^2 + x - 2} = \frac{A}{x-1} + \frac{B}{x+2} $$

Where we need to find the values of A and B. Let's dive into how we do this!

Key Concepts and Steps

  1. Proper vs. Improper Rational Functions: First, we need to make sure our rational function is "proper". A rational function is proper if the degree of the numerator, $P(x)$, is *less than* the degree of the denominator, $Q(x)$. If it's not (i.e., it's improper), we need to perform polynomial long division *first* until we get a remainder $R(x)$ where the degree of $R(x)$ is less than the degree of $Q(x)$. This gives us an expression like:
    2. $$ \frac{P(x)}{Q(x)} = S(x) + \frac{R(x)}{Q(x)} $$

    Where $S(x)$ is the result of the division. The good news is we can integrate $S(x)$ as a polynomial and then focus on dealing with the proper rational function $ \frac{R(x)}{Q(x)} $

  2. Factor the Denominator: The next crucial step is to factor the denominator, $Q(x)$, as much as possible. We want to break it down into linear factors (like $ax + b$) and irreducible quadratic factors (like $ax^2 + bx + c$ where $b^2 - 4ac < 0$). For example: $Q(x) = x^2 - 16 = (x-4)(x+4)$
  3. Partial Fraction Decomposition: Now comes the heart of the method. Based on the factors of $Q(x)$, we express the proper rational function as a sum of partial fractions. There are a few cases to consider here:
    • Case I: Distinct Linear Factors: If $Q(x)$ is a product of distinct linear factors, i.e., $Q(x) = (a_1x + b_1)(a_2x + b_2)...(a_kx + b_k)$, then we can write: $$ \frac{R(x)}{Q(x)} = \frac{A_1}{a_1x + b_1} + \frac{A_2}{a_2x + b_2} + ... + \frac{A_k}{a_kx + b_k} $$
    • Case II: Repeated Linear Factors: If $Q(x)$ has repeated linear factors, i.e., $(a_1x + b_1)^r$ occurs in the factorization of $Q(x)$, then we include terms like: $$ \frac{A_1}{a_1x + b_1} + \frac{A_2}{(a_1x + b_1)^2} + ... + \frac{A_r}{(a_1x + b_1)^r} $$
    • Case III: Irreducible Quadratic Factors: If $Q(x)$ contains irreducible quadratic factors (that can't be factored further), we include terms like: $$ \frac{Ax + B}{ax^2 + bx + c} $$
    • Case IV: Repeated Irreducible Quadratic Factors: For repeated irreducible quadratic factors, $(ax^2 + bx + c)^r$, we have terms like: $$ \frac{A_1x + B_1}{ax^2 + bx + c} + \frac{A_2x + B_2}{(ax^2 + bx + c)^2} + ... + \frac{A_rx + B_r}{(ax^2 + bx + c)^r} $$
  4. Determine the Constants: After setting up the partial fraction decomposition, we need to solve for the unknown constants (the $A$s and $B$s). We can do this by multiplying both sides of the equation by $Q(x)$ and then either:
    • Equating Coefficients: Expanding the right side and equating the coefficients of corresponding terms on both sides of the equation, leading to a system of linear equations.
    • Strategic Substitution: Substituting carefully chosen values of $x$ that simplify the equation and allow us to solve for the constants more easily.
  5. Integrate! Once we have the values of the constants, we can integrate each term in the partial fraction decomposition. These integrals should be much simpler than the original integral!

Don't worry if this seems like a lot at first. We'll work through plenty of examples in class. Remember, practice makes perfect! So keep practicing, and you'll become a partial fraction integration pro in no time. Good luck, and see you in our next zoom meeting!