Ch. 8 - Integration Techniques

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 8 - Integration Techniques

Problem 8.3.68

Chapter 8, Problem 8.3.68

67-70. Integrals of the form ∫ sin(mx)cos(nx) dx Use the following product-to-sum identities to evaluate the given integrals:
sin(mx)sin(nx) = ½[cos((m-n)x) - cos((m+n)x)]
sin(mx)cos(nx) = ½[sin((m-n)x) + sin((m+n)x)]
cos(mx)cos(nx) = ½[cos((m-n)x) + cos((m+n)x)]
68. ∫ sin(5x)sin(7x) dx

Verified step by step guidance

Step 1: Recognize that the integral involves the product of two sine functions, sin(5x) and sin(7x). To simplify this, use the product-to-sum identity for sin(mx)sin(nx): sin(mx)sin(nx) = ½[cos((m-n)x) - cos((m+n)x)].

Step 2: Substitute m = 5 and n = 7 into the identity. This gives sin(5x)sin(7x) = ½[cos((5-7)x) - cos((5+7)x)]. Simplify the expressions inside the cosine functions: (5-7)x = -2x and (5+7)x = 12x.

Step 3: Rewrite the integral using the product-to-sum identity: ∫ sin(5x)sin(7x) dx = ½ ∫ [cos(-2x) - cos(12x)] dx. Note that cos(-2x) = cos(2x) because cosine is an even function.

Step 4: Break the integral into two separate integrals: ∫ sin(5x)sin(7x) dx = ½ [∫ cos(2x) dx - ∫ cos(12x) dx].

Step 5: Integrate each term separately. Recall that the integral of cos(kx) is (1/k)sin(kx) + C, where k is a constant. Apply this formula to both ∫ cos(2x) dx and ∫ cos(12x) dx, and combine the results to complete the solution.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

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

Product-to-Sum Identities

Product-to-sum identities are trigonometric formulas that express products of sine and cosine functions as sums or differences of sine and cosine functions. These identities simplify the integration of products of trigonometric functions by transforming them into a more manageable form, making it easier to evaluate integrals.

Integration Techniques

Integration techniques refer to various methods used to compute integrals, including substitution, integration by parts, and the use of trigonometric identities. In the context of the given question, applying product-to-sum identities is a specific technique that allows for the simplification of integrals involving products of sine and cosine functions.

Definite vs. Indefinite Integrals

Definite integrals calculate the area under a curve between two specified limits, while indefinite integrals represent a family of functions and include a constant of integration. Understanding the difference is crucial when evaluating integrals, as it affects the final result and the interpretation of the integral's meaning in a given context.

Recommended video:

05:43

Definition of the Definite Integral

Related Practice

Textbook Question

7–58. Improper integrals Evaluate the following integrals or state that they diverge.

22. ∫ (from -∞ to -2) (1/x²) sin(π/2) dx

views

Textbook Question

60–69. Completing the square Evaluate the following integrals.

65. ∫[1/2 to (√2 + 3)/(2√2)] dx / (8x² - 8x + 11)

views

Textbook Question

7–40. Table look-up integrals Use a table of integrals to evaluate the following indefinite integrals. Some of the integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table.

18. ∫ dx / (225 − 16x²)

views

Textbook Question

7–84. Evaluate the following integrals.

9. ∫ from 4 to 6 [1 / √(8x – x²)] dx

views

Textbook Question

9–40. Integration by parts Evaluate the following integrals using integration by parts.

14. ∫ s · e⁻²ˢ ds

views

Textbook Question

9–61. Trigonometric integrals Evaluate the following integrals.