Chapter 7 Test Solutions

Welcome to the solutions for the Chapter 7 test! We'll walk through each problem step-by-step, highlighting key concepts and techniques. Remember, understanding the process is just as important as getting the right answer. Don't be afraid to review the integration techniques if you get stuck, and always double-check your work!

Problem 1: Integration by Parts

Let's start with a classic: integration by parts. This technique is essential for integrating products of functions.

Here's the integral: $$\int (x-1) \cos(\pi x) dx$$

We use the formula $\int u dv = uv - \int v du$. Let's choose:

  • $u = x-1$, so $du = dx$
  • $dv = \cos(\pi x) dx$, so $v = \frac{1}{\pi} \sin(\pi x)$

Now, applying the formula:

$$\int (x-1) \cos(\pi x) dx = (x-1)\left(\frac{1}{\pi} \sin(\pi x)\right) - \int \frac{1}{\pi} \sin(\pi x) dx$$

$$= \frac{1}{\pi}(x-1)\sin(\pi x) + \frac{1}{\pi^2} \cos(\pi x) + C$$

Problem 2: Trigonometric Integrals

Next up, we have a trigonometric integral. These often require clever use of trigonometric identities.

$$\int \tan^2(x) \sec^4(x) dx$$

Rewrite as: $$\int \tan^2(x) \sec^2(x) \sec^2(x) dx = \int \tan^2(x) (1 + \tan^2(x)) \sec^2(x) dx$$

Let $u = \tan(x)$, so $du = \sec^2(x) dx$. Then we have:

$$\int u^2 (1 + u^2) du = \int (u^2 + u^4) du = \frac{1}{3}u^3 + \frac{1}{5}u^5 + C$$

$$= \frac{1}{3}\tan^3(x) + \frac{1}{5}\tan^5(x) + C$$

Problem 3: Partial Fraction Decomposition

Partial fraction decomposition is crucial for integrating rational functions.

$$\int \frac{x(3-5x)}{(3x-1)(x-1)^2} dx$$

We decompose the fraction as follows:

$$\frac{x(3-5x)}{(3x-1)(x-1)^2} = \frac{A}{3x-1} + \frac{B}{x-1} + \frac{C}{(x-1)^2}$$

General Tips for Integration

  • Master the Basics: Know your basic integration rules and trigonometric identities inside and out.
  • Practice, Practice, Practice: The more you practice, the better you'll become at recognizing patterns and choosing the right techniques.
  • Don't Give Up: Integration can be challenging, but with persistence, you can master it.

Keep up the great work, and remember that every problem you solve brings you one step closer to mastering calculus!