Section 7-4: Class Notes for 2-9-2023

Welcome to a review of Chapters 6 and 7! Today's focus is to get you ready for the test. We'll be going over some essential integration techniques and volume calculations. Remember, practice is key to mastering these concepts, so let's work through these examples together.

Indefinite Integrals and Substitution

Let's start with a quick review of indefinite integrals. One of the most basic integrals is:

$$\int \frac{1}{x} dx = ln|x| + C$$

A slightly more complex case involves a simple substitution. For example:

$$\int \frac{1}{x+2} dx = ln|x+2| + C$$

Here, we can use the substitution $u = x + 2$, so $du = dx$. This makes the integral straightforward. Don't forget the constant of integration, C!

Partial Fraction Decomposition

Sometimes, we encounter integrals that look intimidating but can be simplified using partial fraction decomposition. Consider this integral:

$$\int \frac{x+5}{x^2+x-2} dx$$

First, factor the denominator: $x^2 + x - 2 = (x-1)(x+2)$. Then, we decompose the fraction:

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

Solving for A and B, we get $A = 2$ and $B = -1$. So, the integral becomes:

$$\int \frac{2}{x-1} - \frac{1}{x+2} dx = 2ln|x-1| - ln|x+2| + C$$

Key Idea: Break down complex rational functions into simpler fractions that are easier to integrate!

General Cases of Partial Fraction Decomposition

Here's a summary of the cases for partial fraction decomposition:

  • Case 1: Distinct Linear Factors

    If $Q(x) = (x + b_1)(x + b_2)...(x + b_k)$, then:

    $$\frac{R(x)}{Q(x)} = \frac{A_1}{x+b_1} + \frac{A_2}{x+b_2} + ... + \frac{A_k}{x+b_k}$$
  • Case 2: Repeated Linear Factors

    If we have repeated factors like $(x + b_1)^r$, then:

    $$\frac{A_1}{x+b_1} + \frac{A_2}{(x+b_1)^2} + ... + \frac{A_r}{(x+b_1)^r}$$
  • Case 3: Irreducible Quadratic Factors

    For factors like $(x^2 + a)$, we use linear terms in the numerator:

    $$\frac{Ax + B}{x^2 + a}$$

Chapter 6 and 7 Test Review Problems

Here are a few example problems that might show up on the test:

  1. Find the volume of the rotated region bounded by the curves $x = y^2$ and $x = 1 - y^2$ about the line $x = -1$.
  2. Evaluate the integral: $\int x^4 ln(x) dx$. (This requires integration by parts!)
  3. Evaluate the definite integral: $\int_{1}^{2} \frac{3x^2+6x+2}{x^2+3x+2} dx$

These problems cover a range of techniques including volume calculation and integration by parts. Make sure to review those methods!

Keep practicing, and you'll be well-prepared for the test. Good luck, and remember, Professor Baker is here to help!