Section 7-4: Integration of Rational Functions by Partial Fractions
Welcome back to Professor Baker's Math Class! In this lesson, we'll explore a powerful technique for integrating rational functions: partial fraction decomposition. Get ready to dust off your algebra skills, because this method is a blend of algebraic manipulation and integral calculus. Don't worry; we'll guide you every step of the way!
What are Rational Functions?
A rational function is simply a ratio of two polynomials, expressed as:
$$f(x) = \frac{P(x)}{Q(x)}$$where $P(x)$ and $Q(x)$ are both polynomials.
The Goal: Simpler Fractions
The main idea behind partial fraction decomposition is to break down a complex rational function into a sum of simpler fractions that we already know how to integrate. For instance, consider:
$$\frac{x+5}{x^2 + x - 2} = \frac{2}{x-1} - \frac{1}{x+2}$$Integrating the left side directly might be challenging, but integrating the right side is straightforward:
$$\int \frac{x+5}{x^2 + x - 2} dx = \int \left( \frac{2}{x-1} - \frac{1}{x+2} \right) dx = 2\ln|x-1| - \ln|x+2| + C$$When to Use Partial Fractions
Partial fraction decomposition is most effective when:
- We are trying to integrate a rational function.
- Direct integration is difficult or impossible.
Steps for Partial Fraction Decomposition
- Check if the rational function is proper: A rational function $\frac{P(x)}{Q(x)}$ is proper if the degree of $P(x)$ is less than the degree of $Q(x)$. If it's not (improper), perform long division until you get a proper rational function plus a polynomial.
- Factor the denominator: Factor $Q(x)$ as much as possible into linear factors (of the form $ax + b$) and irreducible quadratic factors (of the form $ax^2 + bx + c$ where $b^2 - 4ac < 0$).
- Set up the partial fraction decomposition: Based on the factors in the denominator, set up the appropriate form for the decomposition. Here are the cases:
- Distinct Linear Factors: If $Q(x) = (a_1x + b_1)(a_2x + b_2)...(a_kx + b_k)$, then $$\frac{R(x)}{Q(x)} = \frac{A_1}{a_1x + b_1} + \frac{A_2}{a_2x + b_2} + ... + \frac{A_k}{a_kx + b_k}$$
- Repeated Linear Factors: If $Q(x)$ has a repeated factor $(ax + b)^r$, then include the terms $$\frac{A_1}{ax + b} + \frac{A_2}{(ax + b)^2} + ... + \frac{A_r}{(ax + b)^r}$$
- Irreducible Quadratic Factors: If $Q(x)$ has a factor $ax^2 + bx + c$, then include a term of the form $$\frac{Ax + B}{ax^2 + bx + c}$$
- Repeated Irreducible Quadratic Factors: If $Q(x)$ has a repeated factor $(ax^2 + bx + c)^r$, then include the terms $$\frac{A_1x + B_1}{ax^2 + bx + c} + \frac{A_2x + B_2}{(ax^2 + bx + c)^2} + ... + \frac{A_rx + B_r}{(ax^2 + bx + c)^r}$$
- Solve for the constants: Multiply both sides of the equation by $Q(x)$ to clear the denominators. Then, solve for the unknown constants (A, B, C, etc.) by either:
- Equating coefficients of like terms.
- Substituting convenient values of $x$.
- Integrate: Integrate the resulting simpler fractions. You may need to use u-substitution or complete the square for some terms.
Example: Distinct Linear Factors
Let's integrate $\int \frac{x^2 + 2x - 1}{x(2x-1)(x+2)} dx$.
First, we decompose the fraction:
$$\frac{x^2 + 2x - 1}{x(2x-1)(x+2)} = \frac{A}{x} + \frac{B}{2x-1} + \frac{C}{x+2}$$Solving for $A$, $B$, and $C$ (using either method above), we find $A = 1/2$, $B=1/10$, and $C= -1/5$. Therefore:
$$\int \frac{x^2 + 2x - 1}{x(2x-1)(x+2)} dx = \int \left( \frac{1/2}{x} + \frac{1/5}{2x-1} - \frac{1/10}{x+2} \right) dx = \frac{1}{2}\ln|x| + \frac{1}{10}\ln|2x-1| - \frac{1}{5}\ln|x+2| + K$$Keep Practicing!
Partial fraction decomposition can be tricky at first, but with practice, you'll become a pro. Remember to carefully factor the denominator, set up the correct form for the decomposition, and solve for the constants. Good luck, and see you in the next lesson!