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Ch. 5 - Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 5 - IntegrationProblem 5.5.43
Chapter 5, Problem 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.                                                                                  
                                                                                                                                                                    
 βˆ« sin 𝓍 sec⁸ 𝓍 d𝓍

Verified step by step guidance
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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.

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Key Concepts

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

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.
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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.
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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.
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Related Practice
Textbook Question

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

                                                                                                                                                                              

 βˆ«β‚€ΒΉ 2eΒ²Λ£ d𝓍

Textbook Question

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

                                                                                                                                                                    

 βˆ« (𝓍⁢ ― 3𝓍²)⁴ (𝓍⁡ ― 𝓍) d𝓍

Textbook Question

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

                                                                                                                                                                              

 βˆ«β‚€α΅‰Β² (ln p)/p dp

Textbook Question

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? 

Textbook Question

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


4

βˆ‘ Ζ’ (1 + k) β€’ 1 is a right Riemann sum for f on the interval [ ___ , ___ ] with

k = 1

n = ________ .

1
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Textbook Question

Symmetry in integrals Use symmetry to evaluate the following integrals.

βˆ«Β²β‚‹β‚‚ (xΒ² + xΒ³) dx