Problem 57 - Calculus Early Transcendentals: Pearson New International Edition

Question

Some software packages can evaluate indefinite integrals. Use your software on each of the following. (a) \(\int 6 \sin (3(x-2)) d x\) (b) \(\int \sin ^{3}(x / 6) d x\) (c) \(\int\left(x^{2} \cos 2 x+x \sin 2 x\right) d x\)

Step-by-Step Solution

Answer

(a) \(-2 \cos(3(x-2)) + C \); (b) \(-\frac{3}{8} \cos(x/6) - \frac{1}{24} \cos^3(x/6) + C \); (c) \(\frac{x^2}{2} \sin(2x) + \frac{1}{4} x \cos(2x) + \frac{1}{8} \sin(2x) + C\).

Step 1: Set Up the Integral for Evaluation

We are required to evaluate the indefinite integral \( \int 6 \sin (3(x-2)) \, dx \). Represent this integral in mathematical software according to its syntax requirements. Understand that this step only involves preparing the integral for software evaluation without performing manual calculations.

Step 2: Evaluate Integral (a) Using Software

Use the software to evaluate the integral \( \int 6 \sin (3(x-2)) \, dx \). The software will output the antiderivative. For this integral, the result should be: \(-2 \cos(3(x-2)) + C\), where \( C \) is the constant of integration.

Step 3: Set Up the Next Integral

Proceed to integral \( \int \sin^3(x/6) \, dx \). Again, use the software syntax to ensure the correct input of the function. This integral involves a more complex trigonometric power.

Step 4: Evaluate Integral (b) Using Software

Submit \( \int \sin^3(x/6) \, dx \) to the software. The output should be: \(-\frac{3}{8} \cos(x/6) - \frac{1}{24} \cos^3(x/6) + C\). Interpret this as the evaluated integral with the constant \( C\) representing constant integration.

Step 5: Set Up and Evaluate Integral (c)

Enter the integral \( \int (x^2 \cos(2x) + x \sin(2x)) \, dx \) for evaluation. This expression requires the software to apply integration by parts or use advanced integration techniques.

Step 6: Interpret Software Output for Integral (c)

The software will provide an evaluation for \( \int (x^2 \cos(2x) + x \sin(2x)) \, dx \). The antiderivative will be \( \frac{x^2}{2} \sin(2x) + \frac{1}{4} x \cos(2x) + \frac{1}{8} \sin(2x) + C\), where \( C \) is the constant of integration.

Key Concepts

Integration Techniques

When we calculate an indefinite integral, we aim to find a function whose derivative matches the original integrand. It’s like running the process of differentiation in reverse. A variety of techniques exist, but software makes it super easy! Here’s what you typically encounter:

• Basic Integration: This involves simple functions where you can directly find the antiderivative using standard rules, like \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \).
• Substitution Method: Beneficial for integrals with composite functions. For example, the integral \( \int 6 \sin (3(x-2)) \, dx \) requires substitution to simplify inner functions.
• Integration by Parts: Used primarily for products of functions. Our last example involved terms like \( x^2 \cos(2x) \), showcasing when this technique shines.

Using software pushes these techniques into the background, allowing students to see the end result quickly. However, understanding the technique each software uses to reach the conclusion is still essential to grasp the broader concept and apply it manually if needed.

Trigonometric Functions

Trigonometric functions like sine and cosine often pop up in integration problems. They bring unique challenges due to their periodic nature.

• Sine and Cosine Integrals: Typical integrals involve finding antiderivatives like \( \int \sin(ax) \, dx = -\frac{1}{a} \cos(ax) + C \).
• Function Powers: With powers of trigonometric functions, such as \( \sin^3(x/6) \), we often opt for identities to break them down. Using software, it recognizes such identities might run behind the scenes to determine the function's integral.

Understanding how these functions behave and their basic integrals is crucial. Software automatically applies these rules, often complicated by trigonometric identities, to simplify problems faster than traditional means.

Software Evaluation

Utilizing software for evaluating indefinite integrals is increasingly common. This approach automates tedious parts of calculus, allowing students to focus on understanding the big picture.

• Efficiency: Once commands are entered correctly, software can quickly evaluate even complex integrals thanks to its programmed algorithms.
• Understanding the Software: Familiarity with the software's syntax is essential. Errors often arise from incorrect inputs rather than the software itself.
• Constant of Integration: Remember, all indefinite integrals come with a constant \( C \). Software typically includes this to represent an entire family of functions.

Software evaluation for integrals empowers learners by showing immediate results. It helps confirm manual calculations or offer a hint when struggles arise. Yet, being too reliant without understanding could leave you in a bind in testing scenarios or when software isn't available.