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Chapter 6: Problem 22

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

Short Answer

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

Step by step solution

01

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 \(\int \tan(\theta)(\sec^2(\theta)-1)d\theta\)
02

Break up the integral

The next step involves breaking up the integral into two separate integrals:\(\int \tan(\theta) \sec^2(\theta) d \theta - \int \tan(\theta) d \theta\).
03

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 \(-\ln|\cos(\theta)|\)
04

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

Prove that if \(f(x) \geq 0, \lim _{x \rightarrow a} f(x)=0,\) and \(\lim _{x \rightarrow a} g(x)=-\infty,\) then \(\lim _{x \rightarrow a} f(x)^{g(x)}=\infty\)

Laplace Transforms Let \(f(t)\) be a function defined for all positive values of \(t\). The Laplace Transform of \(f(t)\) is defined by \(F(s)=\int_{0}^{\infty} e^{-s t} f(t) d t\) if the improper integral exists. Laplace Transforms are used to solve differential equations. Find the Laplace Transform of the function. $$ f(t)=\cosh a t $$

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(f\) is continuous on \([0, \infty)\) and \(\int_{0}^{\infty} f(x) d x\) diverges, then \(\lim _{x \rightarrow \infty} f(x) \neq 0\)

Evaluate the definite integral. $$ \int_{-\pi}^{\pi} \sin 3 \theta \cos \theta d \theta $$

Find the integral. Use a computer algebra system to confirm your result. $$ \int \frac{\sin ^{2} x-\cos ^{2} x}{\cos x} d x $$
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