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Chapter 7: Problem 28

Evaluate the integral. \( \displaystyle \int \tan^5 x \sec^3 x dx \)

Short Answer

Expert verified
Apply trigonometric identities and integration by parts with \( u = \tan^4(x) \) and \( dv = \sec^3(x) \).

Step by step solution

01

Understand the Integration by Parts Formula

The formula for integration by parts is \( \int u dv = uv - \int v du \). We often use this technique for integrals where the product of two functions is involved, like \( \tan^5 x \sec^3 x \).
02

Choose Appropriate Substitutions

We know \( \tan(x) = \frac{\sin(x)}{\cos(x)} \) and \( \sec(x) = \frac{1}{\cos(x)} \). We generally start with \( u = \tan^n(x) \) and \( dv = \sec^3(x) \). However, due to complexity, let's rewrite \( \tan^5(x) \sec^3(x) \) as \( \tan^4(x) \tan(x) \sec^3(x) \) and set \( u = \tan^4(x) \) and \( dv = \tan(x) \sec^3(x) \).
03

Simplify the Integral Using Trigonometric Identities

Split the power of tangent: \( \tan^4(x) = (\sec^2(x) - 1)^2 \). This transforms the integral to: \[ \int (\sec^2(x) - 1)^2 \tan(x) \sec^3(x) \ dx \]. Expand this and integrate term by term.
04

Apply the Simplified Integral

This yields: \( \int (\sec^4(x) - 2\sec^2(x) + 1) \tan(x) \sec^3(x) \ dx \) which will separate into simpler integrals. Use \( \sec^2(x) = 1 + \tan^2(x) \) to simplify each term as needed.
05

Simplify and Solve Each Integral

The integral now becomes multiple parts: \( \int \sec^7(x) \tan(x) dx - 2 \int \sec^5(x) \tan(x) dx + \int \sec^3(x) \tan(x) dx \). Each part can be simplified using trigonometric identities and integration by parts as needed.

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Key Concepts

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

Trigonometric Identities
To solve integrals, especially ones involving trigonometric functions, it's essential to use trigonometric identities. These identities can help simplify complex expressions and make them more manageable. Here's how they work in the context of integration:
  • Pythagorean Identities: These are among the most commonly used. For example, \( \sec^2(x) = 1 + \tan^2(x) \). This relation is frequently useful in integration to manipulate terms.
  • Quotient and Reciprocal Identities: Like \( \tan(x) = \frac{\sin(x)}{\cos(x)} \) and \( \sec(x) = \frac{1}{\cos(x)} \), these identities simplify functions into basic sine and cosine forms.
  • Power Reduction Identities: Allow breaking down higher powers of trigonometric functions. For instance, using \( \tan^n(x) = (\sec^2(x) - 1)^{\frac{n}{2}} \) helps in manipulating the expression strategically.
Mastering these identities is crucial because they transform complex polynomial forms of trigonometric functions into more integral-friendly formats.
Integration Techniques
Integration techniques are strategies used to find the integral of a function. Here are some key techniques relevant to the given problem:
  • Integration by Parts: This technique is useful when you have the product of two different functions. It relies on the formula \( \int u \, dv = uv - \int v \, du \). Choosing \( u \) and \( dv \) effectively is critical. In our example, we selected \( u = \tan^4(x) \) to exploit its manipulation into simpler terms.
  • Substitution: Simplifying expressions using substitutions helps when performing integration by reducing an expression into a standard integral form.
  • Heuristic Simplification: Often, breaking down complex expressions into sums or differences of simpler integrals can simplify calculation. For example, rewriting \( \tan^5(x) \sec^3(x) \) as \( \tan^4(x) \tan(x) \sec^3(x) \) helps.
Developing a flexible approach to these techniques can greatly simplify the integration process and lead to successful solutions.
Definite and Indefinite Integrals
Integrals can be definite or indefinite, and understanding their application is key in calculus:
  • Indefinite Integrals: They represent functions \( F(x) \) such that \( F'(x) = f(x) \). These integrals have a "+ C" because indefinite integrals find the general form of antiderivatives, accounting for any constant.
  • Definite Integrals: Given by \( \int_a^b f(x) \, dx \), they calculate the net area under the curve from \( a \) to \( b \). They don't need a constant of integration, because it's a precise calculation.
  • Application and Solution: For the problem at hand, understanding when to apply definite or indefinite integrals is important. Solving the integral \( \int \tan^5(x) \sec^3(x) \, dx \) initially as an indefinite integral allows forming a general solution, which can later be evaluated over specific bounds if needed.
Mastery of these integrations and when to apply each form provides clarity and skill when solving complex calculus problems.

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Most popular questions from this chapter

Use (a) the Midpoint Rule and (b) Simpson's Rule to approximate the given integral with the specified value of \( n \). (Round your answers to six decimal places.) Compare your results to the actual value to determine the error in each approximation. \( \displaystyle \int_0^\pi x cos x\ dx \) , \( n = 4 \)

If \( f(t) \) is continuous for \( t \ge 0 \), the Laplace transform of \( f \) is the function \( F \) defined by $$ F(s) = \int_0^\infty f(t) e^{-st}\ dt $$ and the domain of \( F \) is the set consisting of all numbers s for which the integral converges. Find the Laplace transforms of the following functions. (a) \( f(t) = 1 \) (b) \( f(t) = e^t \) (c) \( f(t) = t \)

(a) If \( g(x) = \frac{(\sin^2 x)}{x^2} \), use your calculator or computer to make a table of approximate values of \( \displaystyle \int_1^t g(x)\ dx \) for \( t = 2, 5, 10, 100, 1000 \), and \( 10,000 \). Does it appear that \( \displaystyle \int_1^\infty g(x)\ dx \) is convergent? (b) Use the Comparison Theorem with \( f(x) = \frac{1}{x^2} \) to show that \( \displaystyle \int_1^\infty g(x)\ dx \) is convergent. (c) Illustrate part (b) by graphing \( f \) and \( g \) on the same screen for \( 1 \le x \le 10 \). Use your graph to explain intuitively why \( \displaystyle \int_1^\infty g(x)\ dx \) is convergent.

As we saw in Section 3.8, a radioactive substance decays exponentially: The mass at time \( t \) is \( m(t) = m(0)e^{kt} \), where \( m(0) \) is the initial mass and \( k \) is a negative constant. The mean life \( M \) of an atom in the substance is $$ M = -k \int_0^\infty te^{kt}\ dt $$ For the radioactive carbon isotope, \( ^{14} C \), used in radiocarbon dating, the value of \( k \) is \( -0.000121 \). Find the mean life of a \( ^{14} C \) atom.

Determine whether each integral is convergent or divergent. Evaluate those that are convergent. \( \displaystyle \int_1^\infty \frac{1}{(2x + 1)^3}\ dx \)
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