1. Let's start by stating the problem: We want to learn how to integrate rational functions using partial fractions. 2. The key idea is to express a complicated rational function as a sum of simpler fractions, called partial fractions, which are easier to integrate. 3. Suppose we have a rational function $$\frac{P(x)}{Q(x)}$$ where the degree of $$P(x)$$ is less than the degree of $$Q(x)$$, and $$Q(x)$$ can be factored into linear or irreducible quadratic factors. 4. The general form of partial fractions depends on the factorization of $$Q(x)$$: - For distinct linear factors $$(x - a)$$, use terms like $$\frac{A}{x - a}$$. - For repeated linear factors $$(x - a)^n$$, use terms $$\frac{A_1}{x - a} + \frac{A_2}{(x - a)^2} + \cdots + \frac{A_n}{(x - a)^n}$$. - For irreducible quadratic factors $$(ax^2 + bx + c)$$, use terms like $$\frac{Bx + C}{ax^2 + bx + c}$$. 5. Example: Integrate $$\int \frac{3x + 5}{(x - 1)(x + 2)} dx$$. 6. Step 1: Express $$\frac{3x + 5}{(x - 1)(x + 2)} = \frac{A}{x - 1} + \frac{B}{x + 2}$$. 7. Step 2: Multiply both sides by $$(x - 1)(x + 2)$$ to clear denominators: $$3x + 5 = A(x + 2) + B(x - 1)$$. 8. Step 3: Expand the right side: $$3x + 5 = A x + 2A + B x - B$$. 9. Step 4: Group like terms: $$3x + 5 = (A + B) x + (2A - B)$$. 10. Step 5: Equate coefficients of $$x$$ and constants: $$A + B = 3$$ $$2A - B = 5$$. 11. Step 6: Solve the system: Add equations: $$(A + B) + (2A - B) = 3 + 5 \Rightarrow 3A = 8 \Rightarrow A = \frac{8}{3}$$. Substitute back: $$\frac{8}{3} + B = 3 \Rightarrow B = 3 - \frac{8}{3} = \frac{1}{3}$$. 12. Step 7: Rewrite the integral: $$\int \frac{3x + 5}{(x - 1)(x + 2)} dx = \int \frac{8/3}{x - 1} dx + \int \frac{1/3}{x + 2} dx$$. 13. Step 8: Integrate each term: $$\frac{8}{3} \int \frac{1}{x - 1} dx + \frac{1}{3} \int \frac{1}{x + 2} dx = \frac{8}{3} \ln|x - 1| + \frac{1}{3} \ln|x + 2| + C$$. 14. Final answer: $$\boxed{\int \frac{3x + 5}{(x - 1)(x + 2)} dx = \frac{8}{3} \ln|x - 1| + \frac{1}{3} \ln|x + 2| + C}$$. This method can be extended to more complicated denominators by decomposing into appropriate partial fractions and integrating term by term.