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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 8.7.52
Chapter 8, Problem 8.7.52

49–63. {Use of Tech} Integrating with a CAS Use a computer algebra system to evaluate the following integrals. Find both an exact result and an approximate result for each definite integral. Assume a is a positive real number.
52. ∫ from 0 to π/2 of cos⁶x dx

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Step 1: Recognize that the integral involves cos⁶(x), which is a power of a trigonometric function. To simplify, use the power-reduction formula for cosine: cos²(x) = (1 + cos(2x)) / 2. This will help reduce the power of cosine in the integrand.
Step 2: Rewrite cos⁶(x) as (cos²(x))³ and apply the power-reduction formula to cos²(x). Substitute cos²(x) = (1 + cos(2x)) / 2 into the expression, resulting in cos⁶(x) = [(1 + cos(2x)) / 2]³.
Step 3: Expand [(1 + cos(2x)) / 2]³ using the binomial theorem. This will give you terms involving powers of cos(2x). The expanded form will include terms like 1, cos(2x), cos²(2x), and cos³(2x).
Step 4: Integrate each term separately over the interval [0, π/2]. For terms involving cos²(2x) and cos³(2x), use appropriate trigonometric identities or substitution methods to simplify the integration process. For example, cos²(2x) can be reduced using the power-reduction formula again.
Step 5: Use a computer algebra system (CAS) to evaluate the integral for both the exact result and the approximate numerical result. Input the integral ∫ from 0 to π/2 of cos⁶(x) dx into the CAS, and it will compute the values for you.

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Key Concepts

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

Definite Integral

A definite integral represents the signed area under a curve defined by a function over a specific interval. It is calculated using the Fundamental Theorem of Calculus, which connects differentiation and integration. The result of a definite integral is a number that quantifies the accumulation of quantities, such as area, over the interval from the lower limit to the upper limit.
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Computer Algebra System (CAS)

A Computer Algebra System (CAS) is software designed to perform symbolic mathematics, allowing users to manipulate mathematical expressions in a way similar to traditional algebra. CAS can evaluate integrals, solve equations, and simplify expressions, providing both exact and numerical results. This technology is particularly useful for complex integrals that are difficult to solve analytically.
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Trigonometric Integrals

Trigonometric integrals involve the integration of functions that include trigonometric functions, such as sine and cosine. These integrals often require specific techniques, such as substitution or the use of trigonometric identities, to simplify the integrand. In the case of the integral ∫ cos⁶x dx, recognizing patterns and applying reduction formulas can facilitate the evaluation process.
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Related Practice
Textbook Question

9–61. Trigonometric integrals Evaluate the following integrals.

26. ∫ sin³x cos³ᐟ²x dx

Textbook Question

Choosing an integration strategy Identify a technique of integration for evaluating the following integrals. If necessary, explain how to first simplify the integrand before applying the suggested technique of integration. You do not need to evaluate the integrals.

∫ (5x² + 18x + 20) / [(2x + 3)(x² + 4x + 8)] dx

Textbook Question

7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.

36. ∫[8√2 to 16] 1/√(x² - 64) dx

Textbook Question

7–64. Integration review Evaluate the following integrals.

47. ∫ dx / (x⁻¹ + 1)

Textbook Question

58–61. {Use of Tech} Using Simpson's Rule Approximate the following integrals using Simpson's Rule. Experiment with values of n to ensure the error is less than 10⁻³.

60. ∫(from 0 to π) ln(2 + cos x) dx = π ln((2 + √3)/2)

Textbook Question

23-26. {Use of Tech} Simpson's Rule approximations. Find the indicated Simpson's Rule approximations to the following integrals.

24. ∫(4 to 8) √x dx using n = 4 and n = 8 subintervals

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