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Chapter 5: Problem 33

In Exercises \(33-42\) find the indefinite integral. $$\int \cot \frac{\theta}{3} d \theta$$

Short Answer

Expert verified
The indefinite integral of \( \cot(\frac{\theta}{3}) \) is \( 3 \ln | \sin(\frac{\theta}{3}) | \).

Step by step solution

01

Recall Integral of Cotangent

The first step is to recall the integral of cotangent which is \( \int \cot(x) dx = \ln |\sin(x)| \).
02

Substitute

Next, apply a simple substitution for the variable inside the cotangent function. Substitute \( u = \frac{\theta}{3} \), hence \( d\theta = 3du \). Substituting these into the integral, it becomes \( 3 \int \cot (u) du \).
03

Integrate

Now, evaluate the integral: \( 3 \int \cot (u) du = 3 \ln |\sin(u)| \).
04

Back Substitute

Finally, replace \( u \) with \( \frac{\theta}{3} \), from our initial substitution. The integral thus becomes \( 3 \ln| \sin(\frac{\theta}{3}) | \).

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Key Concepts

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

Integration by Substitution
Integration by substitution is a helpful technique, much like solving a puzzle by replacing a complex piece with something simpler. It's often used when a function is wrapped inside another function, making direct integration challenging. This method involves:
  • Identifying a part of the integrand (the function being integrated) that can be substituted with a variable like u.
  • By making a substitution, you simplify the integrand to an easier form.
  • After integrating, remember to replace the substitution with the original variable.
In our example, we chose u = \( \frac{\theta}{3} \), which simplified the cotangent function, enabling us to integrate it smoothly. This simplifies our work, transforming a complex problem into a more approachable one.
Integral of Cotangent
Understanding how to find the integral of cotangent is essential when dealing with trigonometric integrals. The cotangent function, \( \cot(x) \), is the reciprocal of the tangent function and can be expressed through sine and cosine: \( \cot(x) = \frac{\cos(x)}{\sin(x)} \).
This transformation helps us see the integral more clearly. When we integrate \( \cot(x) \), we find that \( \int \cot(x) \, dx \) yields \( \ln |\sin(x)| + C \), where \( C \) is the constant of integration.
To summarize:
  • The integral of the cotangent function is tied to the natural logarithm of the sine function.
  • Careful attention to absolute values is important, especially since sine can be negative or positive.
This knowledge provides a foundation for integrating other trigonometric functions using similar methods.
Inverse Trigonometric Functions
Inverse trigonometric functions are the flip-side to trigonometric functions. They tell us what angle leads to a particular sin, cos, or tan value. Though not directly used in every integral, they're important for understanding the range of values these functions can take and are sometimes involved in results.
For example:
  • \( \arcsin(x) \) gives the angle whose sine is x.
  • \( \arccos(x) \) gives the angle whose cosine is x.
  • \( \arctan(x) \) is used frequently in solving integrals involving tan functions.
Such inverse functions can appear during integration tasks when back substitution yields results in terms of the angles, particularly when the substitution or transformation used involves trigonometric identities. Understanding how these functions work enriches problem-solving skills and aids in recognizing when inverse trigonometric results might occur.

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Most popular questions from this chapter

Modeling Data A valve on a storage tank is opened for 4 hours to release a chemical in a manufacturing process. The flow rate \(R\) (in liters per hour) at time \(t\) (in hours) is given in the table. \(\begin{array}{|c|c|c|c|c|c|}\hline t & {0} & {1} & {2} & {3} & {4} \\ \hline R & {425} & {240} & {118} & {71} & {36} \\ \hline\end{array}\) (a) Use the regression capabilities of a graphing utility to find a linear model for the points \((t, \ln R) .\) Write the resulting equation of the form \(\ln R=a t+b\) in exponential form. (b) Use a graphing utility to plot the data and graph the exponential model. (c) Use a definite integral to approximate the number of liters of chemical released during the 4 hours.

In Exercises \(41-56,\) find the derivative of the function. $$f(x)=\operatorname{arccot} \sqrt{x}$$

Evaluating a Definite Integral In Exercises \(77-80\) , evaluate the definite integral. Use a graphing utility to verify your result. $$\int_{0}^{1}\left(5^{x}-3^{x}\right) d x$$

Consider the function $$F(x)=\frac{1}{2} \int_{x}^{x+2} \frac{2}{t^{2}+1} d t$$ (a) Write a short paragraph giving a geometric interpretation of the function \(F(x)\) relative to the function $$f(x)=\frac{2}{x^{2}+1}$$ Use what you have written to guess the value of \(x\) that will make \(F\) maximum. (b) Perform the specified integration to find an alternative form of \(F(x) .\) Use calculus to locate the value of \(x\) that will make \(F\) maximum and compare the result with your guess in part (a).

The antiderivative of $$\int \frac{1}{\sqrt{1-x^{2}}} d x$$ can be either arcsin \(x+C\) or \(-\arccos x+C .\) Does this mean that arcsin \(x=-\arccos x ?\) Explain.
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