Problem 31
Question
Finding an Indefinite Integral of a Trigonometric Function In Exercises \(31-40\) , find the indefinite integral. $$ \int \cot \frac{\theta}{3} d \theta $$
Step-by-Step Solution
Verified Answer
The indefinite integral of \(\cot(\frac{\theta}{3})\) is \(3 \ln |\sin(\frac{\theta}{3})| + C\), where \(C\) represents the constant of integration.
1Step 1: Substitution
Set \(u= \frac{\theta}{3}\). Then, compute the corresponding differential \(du\) from \(d \theta\) using the chain rule, implying \(3 du = d \theta\).
2Step 2: Rewrite the Integral and Solve
Rewrite the original integral in terms of the new variable, yielding: \(\int \cot u * 3 du\). Now, we integrate. The antiderivative of cotangent is well-known to be \(\ln |\sin u|\), so the result of the integral will be \(3 \ln |\sin u|\).
3Step 3: Undo the Substitution
Return the integral to the variable \(\theta\) using the same substitution from Step 1. The final answer would then be: \(3 \ln |\sin (\frac{\theta}{3})|\).
Key Concepts
Integration TechniquesTrigonometric IntegrationSubstitution Method in IntegrationAntiderivatives
Integration Techniques
When approaching the calculation of indefinite integrals, mastering several integration techniques is crucial for success. These strategies transform complex expressions into forms that are easier to manage and integrate. Common techniques include the substitution method, which is especially useful for integrals involving composite functions, integration by parts for products of functions, and trigonometric integrals, which often require specific identities or transformations. Practice in identifying which technique to apply comes from experience with a variety of integrals and understanding the fundamental concepts behind each method.
Trigonometric Integration
The integration of trigonometric functions can seem daunting due to the myriad of functions and identities involved. The trick is to know the basic antiderivatives of each main trigonometric function, such as \( \sin(x) \), \( \cos(x) \), and \( \tan(x) \). Then, employ strategies like using trigonometric identities to simplify the integral or adopting the substitution method to make it easier to recognize these basic forms.
For instance, with the integral \( \int \cot \frac{\theta}{3} d \theta \), recognizing that \( \cot(x) \) can be expressed as \( \frac{1}{\tan(x)} \) or \( \frac{\cos(x)}{\sin(x)} \) may provide a pathway to find a more straightforward antiderivative. Furthermore, knowing the relationships and derivatives of inverse trigonometric functions expands the toolkit for integrating more complex trigonometric expressions.
For instance, with the integral \( \int \cot \frac{\theta}{3} d \theta \), recognizing that \( \cot(x) \) can be expressed as \( \frac{1}{\tan(x)} \) or \( \frac{\cos(x)}{\sin(x)} \) may provide a pathway to find a more straightforward antiderivative. Furthermore, knowing the relationships and derivatives of inverse trigonometric functions expands the toolkit for integrating more complex trigonometric expressions.
Substitution Method in Integration
The substitution method is akin to the algebraic technique of a variable change, providing a simpler pathway through difficult integrals. It works exceptionally well with composites of functions and when dealing with derivatives that are present in the integrand.
The technique involves choosing a new variable, commonly denoted \( u \), to replace a part of the original integrand. The differential \( du \) is then calculated, often through the chain rule, and the integrand is rewritten in terms of \( u \) and \( du \). This transforms the integral into a more tractable form. After solving the transformed integral, it's important to undo the substitution, converting back to the original variable to complete the solution.
The technique involves choosing a new variable, commonly denoted \( u \), to replace a part of the original integrand. The differential \( du \) is then calculated, often through the chain rule, and the integrand is rewritten in terms of \( u \) and \( du \). This transforms the integral into a more tractable form. After solving the transformed integral, it's important to undo the substitution, converting back to the original variable to complete the solution.
Antiderivatives
An antiderivative of a function is essentially the reverse of taking a derivative: it's a new function whose derivative is the original function. The indefinite integral of a function is equivalent to finding its antiderivative, and because differentiation is a precise operation but integration is more artful, there can be many antiderivatives for any given function, distinguished by a constant of integration, \( C \).
Knowing the standard antiderivatives of basic functions, such as polynomials, exponential functions, and trigonometric functions, is invaluable. For example, knowing that the antiderivative of \( \cot(x) \) is \( \ln |\sin(x)| \) simplifies solving an integral like \( \int \cot \frac{\theta}{3} d \theta \) immensely. The real art of finding antiderivatives lies in recognizing when to apply different techniques, like the substitution method, to reveal these basic forms.
Knowing the standard antiderivatives of basic functions, such as polynomials, exponential functions, and trigonometric functions, is invaluable. For example, knowing that the antiderivative of \( \cot(x) \) is \( \ln |\sin(x)| \) simplifies solving an integral like \( \int \cot \frac{\theta}{3} d \theta \) immensely. The real art of finding antiderivatives lies in recognizing when to apply different techniques, like the substitution method, to reveal these basic forms.
Other exercises in this chapter
Problem 31
Inverse Functions In Exercises \(29-32\) , illustrate that the functions are inverses of each other by graphing both functions on the same set of coordinate axe
View solution Problem 31
In Exercises 29–34, write the expression as a logarithm of a single quantity. $$ \frac{1}{3}\left[2 \ln (x+3)+\ln x-\ln \left(x^{2}-1\right)\right] $$
View solution Problem 31
Show that \(f\) is strictly monotonic on the given interval and therefore has an inverse function on that interval. \(f(x)=\frac{4}{x^{2}}, \quad(0, \infty)\)
View solution Problem 32
In Exercises 23–32, find the derivative of the function. $$ g(x)=\operatorname{sech}^{2} 3 x $$
View solution