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Chapter 8: Problem 45

Use reduction formulas to evaluate the integrals in Exercises \(41-50 .\) $$ \int 4 \tan ^{3} 2 x d x $$

Short Answer

Expert verified
The integral evaluates to \( \tan^2(2x) - \ln|1+\tan^2(2x)| + C \).

Step by step solution

01

Identify Relevant Reduction Formula

For reducing integrals of the form \( \int \tan^{n}(ax)dx \), a common reduction formula is \( \int \tan^{n}(x)dx = \frac{1}{n-1}\tan^{n-1}(x) - \int \tan^{n-2}(x)dx \) when \( n > 1 \). We'll utilize a similar concept here.
02

Substitute for \( \tan(2x) \)

Let \( u = \tan(2x) \). Then the derivative \( \frac{du}{dx} = 2 \sec^2(2x) \) implies \( dx = \frac{du}{2 \sec^2(2x)} \). Now the integral becomes \( 4 \int \tan^3(2x) dx = 4 \int u^3 \frac{du}{2\sec^2(2x)} \).
03

Simplify the Integral

Since \( \sec^2(2x) = 1 + u^2 \), substitute this into the integral: \( 4 \int u^3 \frac{du}{2 (1 + u^2)} \). This can be simplified to \( 2 \int \frac{u^3}{1+u^2} du \).
04

Split the Integral Using Polynomial Long Division

Perform polynomial long division on \( \frac{u^3}{1+u^2} \), which results in \( u - u/(1+u^2) \). This transforms the integral to \( 2 \left( \int u \, du - \int \frac{u}{1+u^2} du \right) \).
05

Integrate Each Term Separately

Evaluate \( \int u \, du \), resulting in \( \frac{u^2}{2} \), and \( \int \frac{u}{1+u^2} du = \frac{1}{2} \ln|1+u^2| \).
06

Combine Results and Back-Substitute

Combine the results from previous step: \( 2 \left( \frac{u^2}{2} - \frac{1}{2} \ln|1+u^2| \right) = u^2 - \ln|1+u^2| \). Back-substitute \( u = \tan(2x) \) to get \( \tan^2(2x) - \ln|1+\tan^2(2x)| + C \), 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 Techniques
Integration is a fundamental concept in calculus, providing a way to compute areas under curves and solve various practical problems. There are multiple techniques to tackle integrals, especially when basic methods like substitution and integration by parts aren鈥檛 efficient. One such method is using **reduction formulas**. These formulas transform complex integrals into simpler ones by reducing the power of the integral step-by-step, making them more manageable.
Reduction formulas are valuable in handling integrals of trigonometric functions, particularly when dealing with higher powers of sine, cosine, or tangent. For the exercise involving \( \int 4 \tan^3 2x \, dx \), a specialized reduction formula helps by expressing \( \tan^n(x) \) in terms of lower powers. This simplifies the computation significantly, breaking it down into a sequence of less complicated integrals. Such a structured approach is crucial for mastering integration techniques.
Calculus Exercises
Calculus exercises serve as the practice ground to apply integration techniques and solidify understanding. When dealing with exercises that involve reduction formulas, it鈥檚 important to:
  • Identify the form of the integral, recognizing if it fits well with any known formulas.
  • Break down the integral into simpler components whenever possible, as demonstrated in our solution, simplifying \( \tan^3(2x) \) using polynomial long division.
  • Work carefully through substitutions, as they often convert trigonometric functions into algebraic forms that are easier to integrate.
Practicing these exercises helps to improve problem-solving skills and flexibility in thought, as each problem may require a different approach or combination of techniques. It鈥檚 not just about finding solutions but understanding why one method works better than another.
Trigonometric Integrals
Integrating trigonometric functions can be challenging due to their periodic nature and the complexity of their higher powers. When addressing trigonometric integrals, especially ones like \( \int \tan^3(2x) \, dx \), key considerations include:
  • Understanding the specific characteristics of trigonometric functions, such as \( \tan(x) \), which can be rewritten using identities for easier integration.
  • Knowing when to use trigonometric identities and substitutions to simplify the integral. In our example, recognizing that \( \sec^2(x) = 1 + \tan^2(x) \) was instrumental in simplifying the integral.
  • Using reduction formulas, as they are indispensable for handling integrals of trigonometric functions raised to a power.
Through practice, these integrals become more intuitive. The methodical breakdown and manipulation of trigonometric expressions into integrable forms illustrate the power and elegance of calculus, providing not just solutions, but enhancing our understanding of function behavior.

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Most popular questions from this chapter

In Exercises \(35-68\) , use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer. $$\int_{-\infty}^{\infty} \frac{d x}{e^{x}+e^{-x}}$$

Miles driven A taxicab company in New York City analyzed the daily number of miles driven by each of its drivers. It found the average distance was 200 mi with a standard deviation of 30 \(\mathrm{mi} .\) Assuming a normal distribution, what prediction can we make about the percentage of drivers who will log in either more than 260 mi or less than 170 \(\mathrm{mi} ?\)

Digestion time The digestion time in hours of a fixed amount of food is exponentially distributed with a mean of 1 hour. What is the probability that the food is digested in less than 30 minutes?

$$ \begin{array}{c}{\text { a. Use a CAS to evaluate }} \\ {\int_{0}^{\pi / 2} \frac{\sin ^{n} x}{\sin ^{n} x+\cos ^{n} x} d x} \\ {\text { where } n \text { is an arbitrary positive integer. Does your CAS find }} \\ {\text { the result? }}\end{array} $$$$ \begin{array}{l}{\text { b. In succession, find the integral when } n=1,2,3,5, \text { and } 7 .} \\ {\text { Comment on the complexity of the results. }}\end{array} $$$$ \begin{array}{l}{\text { c. Now substitute } x=(\pi / 2)-u \text { and add the new and old }} \\ {\text { integrals. What is the value of }} \\\ {\int_{0}^{\pi / 2} \frac{\sin ^{n} x}{\sin ^{n} x+\cos ^{n} x} d x ?} \\\ {\text { This exercise illustrates how a little mathematical ingenuity }} \\\ {\text { Solves a problem not immediately amenable to solution by a }} \\\ {\text { CAS. }}\end{array} $$

In Exercises \(27-40\) , use a substitution to change the integral into one you can find in the table. Then evaluate the integral. $$ \int \frac{d t}{\tan t \sqrt{4-\sin ^{2} t}} $$
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