Problem 45
Question
Find the integral. Use a computer algebra system to confirm your result. $$ \int \csc ^{4} \theta d \theta $$
Step-by-Step Solution
Verified Answer
The integral of \(\csc^{4} \theta d \theta\) is \(-\cot \theta - \frac{2}{3}\cot^{3} \theta - \frac{1}{5}\cot^{5} \theta + C\).
1Step 1: Break down the function
Write the integrand \(\csc^{4} \theta\) as \(\csc^{2} \theta * \csc^{2} \theta\), which can ultimately be expressed as \((1 + \cot^{2} \theta)*(1 + \cot^{2} \theta)\), using the trigonometric identity \(1 + \cot^{2} \theta = \csc^{2} \theta\).
2Step 2: Expand and Simplify
Expand the product to get \(1 + 2\cot^{2}\theta + \cot^{4}\theta\). Now, substitute \(u = \cot \theta\) and \(du = -\csc^{2} \theta d\theta\). This substitution changes the integral into \(-\int (1 + 2u^{2} + u^{4}) du\).
3Step 3: Integrate
Perform the integration using standard rules of integration to get \(-u - \frac{2}{3}u^{3} - \frac{1}{5}u^{5} + C\).
4Step 4: Substitute Back
Substitute \(u = \cot\theta\) back into the equation to get the final answer, which is \(-\cot \theta - \frac{2}{3}\cot^{3} \theta - \frac{1}{5}\cot^{5} \theta + C\).
Key Concepts
Trigonometric IdentitiesSubstitution MethodIntegration Techniques
Trigonometric Identities
In calculus, trigonometric identities serve as crucial tools for simplifying complex expressions, especially when dealing with integration involving trigonometric functions. A trigonometric identity is an equation that is true for every value of the variable. In the problem at hand, the identity \( 1 + \cot^{2} \theta = \csc^{2} \theta \) is pivotal. This identity relates the cotangent and cosecant functions, allowing us to express higher powers of the cosecant function in terms of simpler trigonometric expressions.
The step-by-step solution makes use of this identity by rewriting \( \csc^{4} \theta \) as the square of \( \csc^{2} \theta \). This transformation leads to an expression \( (1 + \cot^{2} \theta)^2 \). By simplifying the given function using trigonometric identities, we can set the stage for more manageable integration.
This technique of recognizing and applying trigonometric identities allows us to break down challenging problems into simpler parts. It is a vital skill for calculus students to master to navigate and solve integrals involving trigonometric functions effectively.
The step-by-step solution makes use of this identity by rewriting \( \csc^{4} \theta \) as the square of \( \csc^{2} \theta \). This transformation leads to an expression \( (1 + \cot^{2} \theta)^2 \). By simplifying the given function using trigonometric identities, we can set the stage for more manageable integration.
This technique of recognizing and applying trigonometric identities allows us to break down challenging problems into simpler parts. It is a vital skill for calculus students to master to navigate and solve integrals involving trigonometric functions effectively.
Substitution Method
The substitution method is a powerful technique in calculus used to simplify and solve integrals. It is particularly helpful when you can transform a complex integrand into a more straightforward form. In this exercise, substitution played a key role by utilizing the substitution \( u = \cot \theta \). This choice simplifies the integration process considerably.
After substituting \( u = \cot \theta \), we find that \( du = -\csc^{2} \theta \, d\theta \). This relationship helps transform the integral from the original trigonometric form into a polynomial form, which is often easier to handle. The original integral \( \int \csc^{4} \theta \, d\theta \) becomes \(-\int (1 + 2u^{2} + u^{4}) \, du \) after substitution, which is significantly more approachable.
The essence of the substitution method is finding the right substitution that simplifies the integral and makes it possible to apply standard integration techniques. After performing the integration with respect to \( u \), the final part of the process is to substitute back the original trigonometric function to express the solution in the terms of the original variable \( \theta \). This crucial step ensures that the solution corresponds to the initial problem.
After substituting \( u = \cot \theta \), we find that \( du = -\csc^{2} \theta \, d\theta \). This relationship helps transform the integral from the original trigonometric form into a polynomial form, which is often easier to handle. The original integral \( \int \csc^{4} \theta \, d\theta \) becomes \(-\int (1 + 2u^{2} + u^{4}) \, du \) after substitution, which is significantly more approachable.
The essence of the substitution method is finding the right substitution that simplifies the integral and makes it possible to apply standard integration techniques. After performing the integration with respect to \( u \), the final part of the process is to substitute back the original trigonometric function to express the solution in the terms of the original variable \( \theta \). This crucial step ensures that the solution corresponds to the initial problem.
Integration Techniques
Integration techniques are diverse strategies used to evaluate integrals, which are foundational to calculus. In this problem, the integrals involved are handled using a combination of algebraic manipulation and the substitution method, which are both essential integration techniques.
The initial step simplifies the integrand using trigonometric identities, creating a framework for applying integration rules. Such techniques often require one to have a strong familiarity with standard integrals, such as \( \int u^n \, du = \frac{u^{n+1}}{n+1} + C \) where \( n eq -1 \). In our exercise, performing the integration on \(-\int (1 + 2u^{2} + u^{4}) \, du \) results in \(-u - \frac{2}{3}u^{3} - \frac{1}{5}u^{5} + C\). This demonstrates how polynomial functions, derived from trigonometric identities, are often easier to integrate.
Understanding and applying integration techniques require practice and a grasp of both algebraic manipulation and calculus concepts. These methods allow students to tackle a wide range of integrals, convert complex expressions into simpler ones, and ultimately solve them efficiently.
The initial step simplifies the integrand using trigonometric identities, creating a framework for applying integration rules. Such techniques often require one to have a strong familiarity with standard integrals, such as \( \int u^n \, du = \frac{u^{n+1}}{n+1} + C \) where \( n eq -1 \). In our exercise, performing the integration on \(-\int (1 + 2u^{2} + u^{4}) \, du \) results in \(-u - \frac{2}{3}u^{3} - \frac{1}{5}u^{5} + C\). This demonstrates how polynomial functions, derived from trigonometric identities, are often easier to integrate.
Understanding and applying integration techniques require practice and a grasp of both algebraic manipulation and calculus concepts. These methods allow students to tackle a wide range of integrals, convert complex expressions into simpler ones, and ultimately solve them efficiently.
Other exercises in this chapter
Problem 44
State the method of integration you would use to perform each integration. Explain why you chose that method. Do not integrate. (a) \(\int x \sqrt{x^{2}+1} d x\
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Find or evaluate the integral using substitution first, then using integration by parts. $$ \int \ln \left(x^{2}+1\right) d x $$
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Evaluate the integral \(\int \frac{x}{x^{2}+9} d x\) using (a) \(u\) -substitution and (b) trigonometric substitution. Discuss the results.
View solution Problem 45
In Exercises \(45-48,\) state whether you would use integration by parts to evaluate the integral. If so, identify what you would use for \(u\) and \(d v\). Exp
View solution