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Chapter 7: Problem 30

Evaluate the integral. $$\int \tan ^{5} x \sec ^{4} x d x$$

Short Answer

Expert verified
\( \frac{(\tan x)^6}{6} + \frac{(\tan x)^8}{4} + \frac{(\tan x)^{10}}{10} + C \)

Step by step solution

01

Identify Substitution Candidates

Identify a substitution that can simplify the integral. The expression has both \( \tan \) and \( \sec \) functions. Recognize that the derivative of \( \tan x \) is \( \sec^2 x \). Therefore, let \( u = \tan x \), which implies \( du = \sec^2 x \, dx \).
02

Express Integral in Terms of Substitution

Rewrite the integral in terms of \( u \) and \( du \). First, express the powers of \( \tan x \) and \( \sec x \) in terms of \( u \): \[ \tan^5 x = u^5 \] \[ \sec^4 x = (1 + \tan^2 x)^2 = (1 + u^2)^2 \] Now, rewrite the integral: \[ \int \tan^5 x \sec^4 x \; dx = \int u^5 (1 + u^2)^2 \sec^2 x \; dx \] But, we have \( \sec^2 x \, dx = du \), so the integral becomes: \[ \int u^5 (1 + u^2)^2 \, du \]
03

Expand and Integrate

Expand \((1 + u^2)^2\) to facilitate integration: \[ (1 + u^2)^2 = 1 + 2u^2 + u^4 \] Substitute back into the integral: \[ \int u^5 (1 + 2u^2 + u^4) \, du = \int (u^5 + 2u^7 + u^9) \, du \] Now integrate term by term: \[ \int u^5 \, du = \frac{u^6}{6} \] \[ \int 2u^7 \, du = \frac{2u^8}{8} = \frac{u^8}{4} \] \[ \int u^9 \, du = \frac{u^{10}}{10} \]
04

Combine Results

Combine the results from integrating each term: \[ \frac{u^6}{6} + \frac{u^8}{4} + \frac{u^{10}}{10} + C \] Where \( C \) is the constant of integration.
05

Back-Substitute for Original Variable

Substitute \( u = \tan x \) back into the expression: \[ \frac{(\tan x)^6}{6} + \frac{(\tan x)^8}{4} + \frac{(\tan x)^{10}}{10} + C \] This is the antiderivative in terms of \( x \).

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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 powerful technique that allows us to simplify complex integrals, especially when they involve composite functions. Think of it as a reverse process of the chain rule in differentiation. The goal is to transform a complicated integral into a simpler one by substituting a part of the integral with a new variable.How to Choose a Substitution:
  • Look for a part of the integrand whose derivative is also present in the expression. This helps simplify the integration process.
  • The choice of substitution doesn't have to be immediately clear. Sometimes trying a substitution is necessary to understand if it simplifies the problem.
In our exercise, we noticed that the derivative of \( \tan x \) is \( \sec^2 x \). Thus, substituting \( u = \tan x \) transforms the cumbersome integral into one that involves powers of \( u \) and \( du \). This kind of selection makes it easier to manage and solve.
Trigonometric Integrals
Trigonometric Integrals deal with integrals involving trigonometric functions such as \( \sin x \), \( \cos x \), \( \tan x \), and \( \sec x \). When solving these, it is helpful to use trigonometric identities to transform the integral into a more workable form.Common Techniques:
  • Use identities like \( \sec^2 x = 1 + \tan^2 x \) and \( \sin^2 x + \cos^2 x = 1 \) to express functions in terms of one another.
  • If the integral contains even powers of \( \sec x \), use these identities to reduce the expression to simpler polynomials.
In our problem, recognizing that \( \sec^4 x = (1 + \tan^2 x)^2 \) allowed us to convert the integral into an expression purely in terms of \( \tan x \). This reduces the complexity and aids in direct integration.
Antiderivatives
Finding an antiderivative is the reverse process of differentiation. It involves finding a function whose derivative gives the integrand. Antiderivatives are critical in solving problems involving indefinite integrals, such as in our exercise.Steps in Finding Antiderivatives:
  • Once the substitution is made, we expand and simplify the expression.
  • Integrate each term separately by applying basic integration rules, such as \( \int x^n \; dx = \frac{x^{n+1}}{n+1} \).
In our solution, we expanded \((1 + u^2)^2\) and integrated each resulting term. For example:
  • The term \( \int u^5 \, du \) becomes \( \frac{u^6}{6} \).
  • The recognition of these patterns in polynomials is the key to solving such integrals efficiently.
Finally, we substitute back \( \tan x \) for \( u \) to express the antiderivative in terms of the original variable, completing the integration process.

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Most popular questions from this chapter

Use any method to find the area of the surface generated by revolving the curve about the \(x\) -axis. $$y=1 / x, 1 \leq x \leq 4$$

What is "improper" about an integral over an infinite interval? Explain why Definition 5.5 .1 for \(\int_{a}^{b} f(x) d x\) fails for \(\int_{a}^{+\infty} f(x) d x .\) Discuss a strategy for assigning a value to \(\int_{a}^{+\infty} f(x) d x\).

Use any method to find the volume of the solid generated when the region enclosed by the curves is revolved about the \(y\) -axis. $$y=\sqrt{x-4}, y=0, x=8$$

Determine whether the statement is true or false. Explain your answer. The trigonometric identity $$\sin \alpha \cos \beta=\frac{1}{2}[\sin (\alpha-\beta)+\sin (\alpha+\beta)]$$ is often useful for evaluating integrals of the form \(\int \sin ^{m} x \cos ^{n} x d x\)

Writing For what sort of problems are the integration techniques of substitution and integration by parts "competing" techniques? Describe situations, with examples, where each of these techniques would be preferred over the other.
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