Problem 22 - Essential Calculus

Question

Use integration tables to find the integral. $$ \int \tan ^{3} \theta d \theta $$

Step-by-Step Solution

Answer

The integral of \(\tan^3(\theta)\) is equal to \(\frac{1}{2} \sec^2(\theta) + \ln|\cos(\theta)| + C\).

Step 1: Express in terms of secant

Firstly, place \(\tan^3(\theta)\) in terms of \(\tan(\theta)\) and \(\tan^2(\theta)\), and further express \(\tan^2(\theta)\) in terms of \(\sec^2(\theta)\) and 1. The integral becomes \(\int \tan(\theta)(\sec^2(\theta)-1)d\theta\)

Step 2: Break up the integral

The next step involves breaking up the integral into two separate integrals:\(\int \tan(\theta) \sec^2(\theta) d \theta - \int \tan(\theta) d \theta\).

Step 3: Use integration rules

We now use the rules of integral calculus to solve the equations. The integral of \(\tan(\theta)\sec^2(\theta)\) is \(\frac{1}{2}\sec^2(\theta)\), and the integral of \(\tan(\theta)\) is \(-\ln|\cos(\theta)|\)

Step 4: Write down the final result

Subtracting the second integral from the first, the final result becomes \(\frac{1}{2} \sec^2(\theta) + \ln|\cos(\theta)| + C\), where C is the constant of integration.

Key Concepts

Integration Tables

Integration tables are a very useful tool in calculus. They provide formulas for common integrals, which can help solve complex problems quickly. These tables are particularly effective when dealing with standard functions like powers, exponentials, logarithms, and trigonometric functions.
This is because they offer ready-made solutions, saving you time from having to derive an integral from scratch.

• Look for the function in question in the table.
• Apply the corresponding formula to your problem.
• Remember to consider constants of integration.

In the given exercise, the use of an integration table can streamline the process of finding the integral of \( an^3(\theta)\) by first expressing components in terms of other trigonometric functions.

Trigonometric Integrals

Trigonometric integrals involve integrals of trigonometric functions, like sine, cosine, and tangent. In the exercise, \( an^3(\theta)\) is expressed in terms of \( an(\theta)\) and \( an^2(\theta)\).
Using trigonometric identities, \(\tan^2(\theta)\) can be rewritten as \(\sec^2(\theta) - 1\).This replacement is useful because it simplifies the integral into components that are easier to handle. Often, breaking trigonometric expressions into known identities can help integrate them:

• \(\sin(\theta)^2 + \cos(\theta)^2 = 1\)
• \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\)
• \(\sec(\theta) = \frac{1}{\cos(\theta)}\)

Understanding these identities can deeply aid in solving integrals involving trigonometric functions.

Integration Rules

Integration rules form the backbone of solving integrals. They include basic rules like the power rule, product rule, and specific formulas for known functions. In this context, we use these rules to split the integral of \(\tan^3(\theta)\) into more manageable parts.
The key rules applied in the solution include:

• The integral of \(\tan(\theta)\sec^2(\theta)\) involves recognizing it as a derivative pattern leading to \(\frac{1}{2}\sec^2(\theta)\).
• Solving \(\int \tan(\theta) d \theta\) using the formula, which results in \(-\ln|\cos(\theta)|\).

By combining the results from these different parts, you compile the complete solution. Remember, these rules are your toolkit for dealing with a variety of integration challenges in calculus.