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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.R.9
Chapter 8, Problem 8.R.9

2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.
9. ∫ (from 0 to π/4) cos⁵ 2x sin² 2x dx

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Step 1: Recognize that the integral involves powers of trigonometric functions. Use trigonometric identities to simplify the expression. Specifically, use the identity sin²θ = 1 - cos²θ to rewrite sin² 2x.
Step 2: Rewrite the integral using substitution. Let u = cos 2x, which implies du = -2 sin 2x dx. Adjust the integral accordingly to account for this substitution.
Step 3: Change the limits of integration to match the substitution. When x = 0, u = cos 0 = 1. When x = π/4, u = cos π/2 = 0. Update the integral limits accordingly.
Step 4: Simplify the integral in terms of u. The integral becomes a polynomial in u, which can be integrated using standard techniques for polynomial integration.
Step 5: After integrating, substitute back the original variable x and evaluate the definite integral using the updated limits. This will yield the final result.

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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 evaluate integrals that may not be solvable by basic antiderivatives. Common techniques include substitution, integration by parts, and trigonometric identities. Understanding these methods is crucial for simplifying complex integrals into manageable forms.
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Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables. They can be used to simplify integrals involving sine and cosine functions. For example, using identities like sin²(x) + cos²(x) = 1 can help rewrite integrals in a more solvable format.
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Definite Integrals

Definite integrals represent the area under a curve between two specified limits. They are calculated using the Fundamental Theorem of Calculus, which connects differentiation and integration. Understanding how to evaluate definite integrals is essential for finding exact values of integrals over a given interval.
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Related Practice
Textbook Question

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

38. ∫ (from π/4 to π/2) x csc²x dx

Textbook Question

Explain why or why not. Determine whether the following statements are true and give an explanation or counterexample.

b. To evaluate the integral ∫dx/√(x² − 100) analytically, it is best to use partial fractions.

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Textbook Question

76-81. Table of integrals Use a table of integrals to evaluate the following integrals.

76. ∫ x(2x + 3)⁵ dx

Textbook Question

Explain why or why not. Determine whether the following statements are true and give an explanation or counterexample.

d. ∫2 sin x cos x dx = −(1/2) cos 2x + C.

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Textbook Question

95–98. {Use of Tech} Numerical integration Estimate the following integrals using the Midpoint Rule M(n), the Trapezoidal Rule T(n), and Simpson’s Rule S(n) for the given values of n.

96. ∫ (from 1 to 3) dx/(x³ + x + 1); n = 4

Textbook Question

76-81. Table of integrals Use a table of integrals to evaluate the following integrals.

79. ∫ sec⁵x dx