Skip to main content
Ch. 5 - Integration
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. 5 - IntegrationProblema 5.5.43
Capitolo 5, Problema 5.5.43

Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating.                                                                                  
                                                                                                                                                                    
 ∫ sin 𝓍 sec⁸ 𝓍 d𝓍

Guida verificata passo dopo passo
1
Step 1: Recognize that the integral involves a combination of trigonometric functions, specifically sin(𝓍) and sec⁸(𝓍). To simplify, consider using a substitution method where u = sec(𝓍).
Step 2: Compute the derivative of u = sec(𝓍). Since the derivative of sec(𝓍) is sec(𝓍)tan(𝓍), we can express d𝓍 in terms of du: d𝓍 = du / (sec(𝓍)tan(𝓍)).
Step 3: Rewrite the integral ∫ sin(𝓍) sec⁸(𝓍) d𝓍 using the substitution u = sec(𝓍). Note that sin(𝓍) can be expressed as tan(𝓍)/sec(𝓍), which simplifies the integral further.
Step 4: Substitute all terms in the integral in terms of u. The integral becomes ∫ (tan(𝓍)/sec(𝓍)) sec⁸(𝓍) (du / (sec(𝓍)tan(𝓍))). Simplify the expression by canceling terms where possible.
Step 5: After simplification, the integral reduces to ∫ u⁷ du. Solve this integral using the power rule for integration: ∫ uⁿ du = (uⁿ⁺¹)/(n+1) + C, where C is the constant of integration. Finally, substitute back u = sec(𝓍) to express the result in terms of 𝓍.

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:
2m

Concetti chiave

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

Indefinite Integrals

Indefinite integrals represent a family of functions whose derivative is the integrand. They are expressed with the integral sign and include a constant of integration, typically denoted as 'C'. The process of finding an indefinite integral is also known as antiderivation, and it is essential for solving problems in calculus, particularly in finding areas under curves and solving differential equations.
Video consigliato:
Percorso guidato
05:04
Introduction to Indefinite Integrals

Change of Variables

Change of variables, or substitution, is a technique used in integration to simplify the integrand. By substituting a new variable for a function of the original variable, the integral can often be transformed into a more manageable form. This method is particularly useful when dealing with complex functions or when the integrand can be expressed in terms of a simpler function, making the integration process easier.
Video consigliato:
Percorso guidato
06:35
Changing Geometries

Differentiation Check

Checking work by differentiation involves taking the derivative of the result obtained from an indefinite integral to verify its correctness. This process ensures that the antiderivative found corresponds to the original integrand. If the derivative of the result matches the integrand, it confirms that the integration was performed correctly, providing a valuable verification step in solving calculus problems.
Video consigliato:
05:02
Determining Differentiability Graphically
Pratica correlata
Domanda del libro di testo

Definite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals.                                                                                                                         

                                                                                                                                                                              

 ∫₀¹ 2e²ˣ d𝓍

74
views
Domanda del libro di testo

Use geometry and properties of integrals to evaluate


∫₀¹ (2𝓍 + √(1―𝓍²) + 1) d𝓍

85
views
Domanda del libro di testo

Definite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals.                                                                                                                         

                                                                                                                                                                              

 ∫₀ᵉ² (ln p)/p dp

76
views
Domanda del libro di testo

Cubic zero net area Consider the graph of the cubic y = 𝓍 (𝓍― a) (𝓍― b), where 0 < a < b. Verify that the graph bounds a region above the 𝓍-axis, for 0 < 𝓍 < a , and bounds a region below the 𝓍-axis, for a < 𝓍 < b. What is the relationship between a and b if the areas of these two regions are equal? 

35
views
Domanda del libro di testo

Identifying Riemann sums Fill in the blanks with an interval and a value of n.


4

∑ ƒ (1 + k) • 1 is a right Riemann sum for f on the interval [ ___ , ___ ] with

k = 1

n = ________ .

101
views
Domanda del libro di testo

Symmetry in integrals Use symmetry to evaluate the following integrals.

∫²₋₂ (x² + x³) dx

113
views