Problem 32 - Thomas Calculus

Question

Use the table of integrals at the back of the book to evaluate the integrals. \(\int \frac{\tan ^{-1} x}{x^{2}} d x\)

Step-by-Step Solution

Answer

The integral is \(-\frac{\tan^{-1} x}{x} + C\).

Step 1: Identify the Integral Form

Look through the table of integrals to find a form that matches \( \int \frac{\tan^{-1} x}{x^2} \, dx \). We find that one of the standard integral forms is \( \int \frac{\tan^{-1} x}{x^2} \, dx = -\frac{\tan^{-1} x}{x} + C \), where \( C \) is the integration constant.

Step 2: Substitute and Simplify

Apply the identified integral form: \( \int \frac{\tan^{-1} x}{x^2} \, dx = -\frac{\tan^{-1} x}{x} + C \). This matches the form provided in the table, so there is no additional simplification required.

Key Concepts

Integration Techniques

Integration is a fundamental concept in calculus that helps us find the total accumulation of quantities, such as areas under curves or the total change given a rate of change. Finding integrals is not always straightforward, which is why integration techniques are extremely valuable.

• Basic Techniques: These include techniques like substitution, integration by parts, and partial fraction decomposition. They are foundational methods you learn early on and solve various integrals by simplifying complex expressions.
• Advanced Techniques: Techniques such as trigonometric integrals and inverse trigonometric functions. These require deeper understanding and often involve memorized forms or transformations.

Knowing when and how to apply each technique efficiently transforms complicated problems into simpler tasks. Mastery of these techniques broadens your capability to tackle diverse problems in integral calculus.

Table of Integrals

A table of integrals can be a lifesaver when dealing with complex integrals that don't easily fit into the basic integration methods. This table provides a list of already-solved integrals for a wide variety of functions, helping you match the problem at hand with its solution.

• Quick Reference: The table serves as a quick reference guide that covers common integral forms.
• Standard Forms: It often includes integrals like exponential functions, trigonometric functions, and logarithmic functions, which are frequently encountered in calculus problems.

To use a table of integrals effectively, identify the form of the integral you are trying to solve and locate it in the table. This way, you can bypass lengthy calculations and verify your results, saving time and reducing errors.

Definite vs Indefinite Integrals

Integrals can be categorized into definite and indefinite types depending on the information you need. Understanding the difference between them is key to applying integrals correctly in calculus problems.

• Indefinite Integrals: Represent a family of functions and include a constant of integration \( C \). They are expressed as \( \int f(x) \, dx \), providing a general formula for the antiderivative of \( f(x) \).
• Definite Integrals: Calculate the specific value of the integral over an interval \([a, b]\), resulting in a numerical value rather than a function. They are displayed as \( \int_{a}^{b} f(x) \, dx \) and often used in calculating areas under curves.

Definite integrals require adding the limits of integration, whereas indefinite integrals result in functions that describe a range of possible values. Each has its applications and uses in solving real-world mathematical and engineering problems.