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Ch. 5 - Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 5 - IntegrationProblem 5.5.90
Chapter 5, Problem 5.5.90

Integrals with sin² 𝓍 and cos² 𝓍 Evaluate the following integrals.                                                                                                             
                                                                                                                                                                    
 ∫₀^π/⁴ cos² 8θ dθ

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Step 1: Recognize that the integral involves cos²(8θ). To simplify this, use the trigonometric identity cos²(x) = (1 + cos(2x)) / 2.
Step 2: Substitute the identity into the integral. The integral becomes ∫₀^(π/4) [(1 + cos(16θ)) / 2] dθ.
Step 3: Split the integral into two separate integrals: (1/2) ∫₀^(π/4) 1 dθ + (1/2) ∫₀^(π/4) cos(16θ) dθ.
Step 4: Evaluate the first integral, (1/2) ∫₀^(π/4) 1 dθ, which is straightforward as it represents the area under a constant function. For the second integral, (1/2) ∫₀^(π/4) cos(16θ) dθ, use the formula for the integral of cos(kx), which is (1/k) sin(kx).
Step 5: Apply the limits of integration (0 to π/4) to both parts of the integral. For the first part, calculate the result of (1/2) θ evaluated at the limits. For the second part, calculate (1/2) * (1/16) * sin(16θ) evaluated at the limits.

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Key Concepts

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

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables. Key identities include the Pythagorean identities, such as sin²(x) + cos²(x) = 1, and double angle formulas. These identities are essential for simplifying integrals involving sin²(x) and cos²(x), allowing for easier evaluation.
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Integration Techniques

Integration techniques are methods used to find the integral of a function. Common techniques include substitution, integration by parts, and using trigonometric identities to simplify the integrand. For integrals involving cos²(θ), applying the identity cos²(θ) = (1 + cos(2θ))/2 can transform the integral into a more manageable form.
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Definite Integrals

Definite integrals calculate the area under a curve between two specified limits. The notation ∫ₐᵇ f(x) dx represents the integral of f(x) from a to b. Evaluating definite integrals often involves finding the antiderivative of the function and applying the Fundamental Theorem of Calculus, which states that the definite integral can be computed by evaluating the antiderivative at the upper and lower limits.
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Related Practice
Textbook Question

Max/min of area functions Suppose ƒ is continuous on [0 ,∞) and A(𝓍) is the net area of the region bounded by the graph of ƒ and the t-axis on [0, x]. Show that the local maxima and minima of A occur at the zeros of ƒ. Verify this fact with the function ƒ(𝓍) = 𝓍² - 10𝓍.

Textbook Question

{Use of Tech} Sigma notation for Riemann sums Use sigma notation to write the following Riemann sums. Then evaluate each Riemann sum using Theorem 5.1 or a calculator.

The right Riemann sum for ƒ(𝓍)) = x + 1 on [0, 4] with n = 50.

Textbook Question

Definite integrals Use geometry (not Riemann sums) to evaluate the following definite integrals. Sketch a graph of the integrand, show the region in question, and interpret your result.                                                                                                                                      

                                                                                                                                                                                       

 ∫₀⁴ (8―2𝓍) d𝓍

Textbook Question

Variations on the substitution method Evaluate the following integrals.                                                                                                        

                                                                                                                                                                    

 ∫ (eˣ ― e⁻ˣ)/ (eˣ + e⁻ˣ) d𝓍

Textbook Question

Average values Find the average value of the following functions on the given interval. Draw a graph of the function and indicate the average value.

ƒ(𝓍) = 1/(𝓍² + 1) on [―1, 1]

Textbook Question

Areas of regions Find the area of the following regions.                                                                                                                   

                                                                                                                                                                 The region bounded by the graph of ƒ(𝓍) = (𝓍―4)⁴ and the 𝓍-axis between and 𝓍 = 2 and 𝓍= 6

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