Skip to main content
Ch. 8 - Integration Techniques
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 8 - Integration TechniquesProblema 8.3.16
Capitolo 8, Problema 8.3.16

9–61. Trigonometric integrals Evaluate the following integrals.
16. ∫ sin²θ cos⁵θ dθ

Guida verificata passo dopo passo
1
Step 1: Recognize that the integral involves powers of sine and cosine. To simplify, use trigonometric identities. Specifically, use the Pythagorean identity: sin²θ = 1 - cos²θ.
Step 2: Rewrite the integral by substituting sin²θ with 1 - cos²θ. The integral becomes: ∫ (1 - cos²θ) cos⁵θ dθ.
Step 3: Expand the expression inside the integral. Distribute cos⁵θ to both terms, resulting in: ∫ cos⁵θ dθ - ∫ cos⁷θ dθ.
Step 4: Use the substitution method for each term. Let u = cosθ, then du = -sinθ dθ. Rewrite the integral in terms of u.
Step 5: Solve each integral separately in terms of u, then substitute back cosθ to express the solution in terms of θ. Combine the results to obtain the final expression.

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.
Durata del video:
5m

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables. Key identities include the Pythagorean identities, angle sum and difference identities, and double angle formulas. These identities can simplify integrals involving trigonometric functions, making it easier to evaluate them.
Video consigliato:
7:17
Verifying Trig Equations as Identities

Integration Techniques

Integration techniques are methods used to find the integral of a function. Common techniques include substitution, integration by parts, and trigonometric substitution. For integrals involving products of trigonometric functions, such as sin²θ and cos⁵θ, using substitution or recognizing patterns can significantly simplify the process.
Video consigliato:
Percorso guidato
06:18
Integration by Parts for Definite Integrals

Power Reduction Formulas

Power reduction formulas are used to express higher powers of sine and cosine in terms of first powers. For example, sin²θ can be rewritten using the identity sin²θ = (1 - cos(2θ))/2. This technique is particularly useful in integrals where the powers of trigonometric functions are high, allowing for easier integration.
Video consigliato:
05:58
Intro to Power Series
Pratica correlata
Domanda del libro di testo

63. (Use of Tech) Normal distribution of heights

The heights of U.S. men are normally distributed with a mean of 69 in and a standard deviation of 3 in. This means that the fraction of men with a height between a and b (with a < b) inches is given by the integral

(1/(3√(2π))) ∫ₐᵇ e^(-((x-69)/3)²/2) dx.

What percentage of American men are between 66 and 72 inches tall? Use the method of your choice, and experiment with the number of subintervals until you obtain successive approximations that differ by less than 10⁻³.

74
views
Domanda del libro di testo

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

51. ∫ x²/√(4 + x²) dx

50
views
Domanda del libro di testo

29-34. {Use of Tech} Comparing the Midpoint and Trapezoid Rules

Apply the Midpoint and Trapezoid Rules to the following integrals. Make a table similar to Table 8.5 showing the approximations and errors for n = 4, 8, 16, and 32. The exact values of the integrals are given for computing the error.

33. ∫(0 to π) sin x cos(3x) dx = 0

70
views
Domanda del libro di testo

7–84. Evaluate the following integrals.

33. ∫ [eˣ / (a² + e²ˣ)] dx, where a ≠ 0

61
views
Domanda del libro di testo

76–83. Preliminary steps The following integrals require a preliminary step such as a change of variables before using the method of partial fractions. Evaluate these integrals.

82. ∫ [dx / (x√(1 + 2x))]

87
views
Domanda del libro di testo

7–64. Integration review Evaluate the following integrals.

12. ∫ from -5 to 0 of dx / √(4 - x)

48
views