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Ch. 8 - Integration Techniques
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 8 - Integration TechniquesProblem 8.RE.6
Chapter 8, Problem 8.RE.6

2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.
6. ∫ (2 − sin 2θ)/cos² 2θ dθ

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1
Rewrite the integral by separating the terms in the numerator: ∫ (2 - sin(2θ)) / cos²(2θ) dθ = ∫ (2 / cos²(2θ)) dθ - ∫ (sin(2θ) / cos²(2θ)) dθ.
For the first term, recognize that 2 / cos²(2θ) can be rewritten using the secant function: 2 / cos²(2θ) = 2 sec²(2θ). The integral of sec²(x) is a standard result, so ∫ 2 sec²(2θ) dθ = (1/2) tan(2θ) + C.
For the second term, rewrite sin(2θ) / cos²(2θ) as (sin(2θ) / cos(2θ)) * (1 / cos(2θ)). Notice that (sin(2θ) / cos(2θ)) is equivalent to tan(2θ), and 1 / cos(2θ) is sec(2θ). Thus, the term becomes ∫ tan(2θ) sec(2θ) dθ.
To integrate ∫ tan(2θ) sec(2θ) dθ, use the substitution u = cos(2θ), which implies du = -2 sin(2θ) dθ. Rewrite the integral in terms of u and solve.
Combine the results of the two integrals, simplify, and include the constant of integration, C, to express the final solution.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Integration Techniques

Integration techniques are methods used to find the integral of a function. Common techniques include substitution, integration by parts, and partial fraction decomposition. Understanding these methods is crucial for evaluating complex integrals, as they allow for simplification and manipulation of the integrand to make integration feasible.
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Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the variables. Key identities, such as the Pythagorean identities and angle sum formulas, can simplify integrals involving trigonometric functions. Recognizing and applying these identities is essential for transforming the integrand into a more manageable form.
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Substitution Method

The substitution method is a technique used in integration where a new variable is introduced to simplify the integral. By substituting a part of the integrand with a single variable, the integral can often be transformed into a standard form that is easier to evaluate. This method is particularly useful when dealing with composite functions or when the integrand contains a function and its derivative.
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Related Practice
Textbook Question

122. Comparing areas The region R₁ is bounded by the graph of y = tan(x) and the x-axis on the interval [0, π/3].

The region R₂ is bounded by the graph of y = sec(x) and the x-axis on the interval [0, π/6]. Which region has the greater area?

Textbook Question

2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.

63. ∫ dx/(x² - 2x - 15)

1
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Textbook Question

2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.

22. ∫ tan³ 5θ dθ

Textbook Question

2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.

51. ∫ (from 0 to π/4) sin⁵(4θ) dθ

Textbook Question

118. Two worthy integrals

b. Let f be any positive continuous function on the interval [0, π/2]. Evaluate

∫ from 0 to π/2 of [f(cos x) / (f(cos x) + f(sin x))] dx.

(Hint: Use the identity cos(π/2 − x) = sin x.)


(Source: Mathematics Magazine 81, 2, Apr 2008)

Textbook Question

2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.

32. ∫ csc²(6x) cot(6x) dx