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 8.7.8

Chapter 8, Problem 8.7.8

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.
8. ∫ sin 3x cos 2x dx

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Step 1: Recognize that the integral ∫ sin(3x) cos(2x) dx involves the product of sine and cosine functions. To simplify, use the trigonometric identity: sin(A)cos(B) = 1/2 [sin(A+B) + sin(A−B)].

Step 2: Apply the identity to rewrite the integral: ∫ sin(3x)cos(2x) dx = 1/2 ∫ [sin(3x+2x) + sin(3x−2x)] dx.

Step 3: Simplify the arguments of the sine functions: 3x + 2x = 5x and 3x − 2x = x. The integral becomes: 1/2 ∫ [sin(5x) + sin(x)] dx.

Step 4: Split the integral into two separate integrals: 1/2 [∫ sin(5x) dx + ∫ sin(x) dx].

Step 5: Use a table of integrals to find the antiderivatives of sin(5x) and sin(x). For sin(kx), the antiderivative is −(1/k)cos(kx) + C. Apply this formula to both terms and combine the results.

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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. For example, the product-to-sum identities can be used to simplify products of sine and cosine functions, such as sin(3x)cos(2x), into a sum of sine or cosine functions, making integration easier.

Integration Techniques

Integration techniques are methods used to evaluate integrals that may not be straightforward. Common techniques include substitution, integration by parts, and using tables of integrals. In this case, recognizing the need to simplify the integral before applying a table is crucial for finding the solution.

Completing the Square

Completing the square is a method used to transform a quadratic expression into a perfect square trinomial. This technique is often necessary when dealing with integrals involving quadratic expressions in the integrand, as it can simplify the integral and make it easier to evaluate using standard integral tables.

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Completing the Square to Rewrite the Integrand

Related Practice

Textbook Question

41–48. Geometry problems Use a table of integrals to solve the following problems.

46. Find the area of the region bounded by the graph of y = 1/√(x² - 2x + 2) and the x-axis from x = 0 to x = 3.

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

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Textbook Question

72. Between the sine and inverse sine Find the area of the region bound by the curves y = sin x and y = sin⁻¹x on the interval [0, 1/2].

Textbook Question

4. Is a reduction formula an analytical method or a numerical method? Explain.

Textbook Question

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

39. ∫ x²/(100 - x²)^(3/2) dx

Textbook Question

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)]

70. ∫ cos(x)cos(2x) dx

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.8. ∫ sin 3x cos 2x dx

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Key Concepts

Trigonometric Identities

Integration Techniques

Completing the Square

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.
8. ∫ sin 3x cos 2x dx