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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 5.5.26
Chapter 5, Problem 5.5.26

Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating.                                                                                  
                                                                                                                                                                    
 βˆ« d𝓍 / (√1 ― 9𝓍²)

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Step 1: Recognize that the integral ∫ d𝓍 / (√1 ― 9𝓍²) resembles the standard form of an integral involving inverse trigonometric functions. Specifically, it matches the form ∫ dx / √(aΒ² - xΒ²), which corresponds to arcsin(x/a) + C.
Step 2: Identify the constants in the given integral. Here, aΒ² = 1, so a = √1 = 1. Additionally, the term 9𝓍² can be rewritten as (3𝓍)Β², which suggests a substitution to simplify the integral.
Step 3: Perform a substitution to simplify the integral. Let u = 3𝓍, which implies that du = 3 d𝓍 or d𝓍 = du / 3. Substitute these into the integral to rewrite it in terms of u.
Step 4: After substitution, the integral becomes (1/3) ∫ du / √(1 - u²). This matches the standard form ∫ dx / √(a² - x²), where a = 1. The result of this integral is (1/3) arcsin(u/a) + C.
Step 5: Substitute back u = 3𝓍 into the result to express the solution in terms of the original variable 𝓍. The final answer is (1/3) arcsin(3𝓍) + C. Verify the solution by differentiating it to ensure it matches the original integrand.

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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 without limits and include a constant of integration, typically denoted as 'C'. The process of finding an indefinite integral is often referred to as antiderivation, and it is fundamental in calculus for solving problems related to area, accumulation, and other applications.
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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 expressions or when the integrand resembles a known derivative.
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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. If the derivative of the antiderivative matches the original integrand, the solution is confirmed to be correct. This step is crucial in calculus as it ensures that the integration process was performed accurately.
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Related Practice
Textbook Question

Identifying definite integrals as limits of sums Consider the following limits of Riemann sums for a function Ζ’ on [a,b]. Identify Ζ’ and express the limit as a definite integral.                                

          n                                                                                                                                                                              

    lim   βˆ‘   π“*β‚– (ln 𝓍*β‚–) βˆ†π“β‚– on [1,2]                                                                                                                                                                            

  βˆ† β†’ 0   k=1                                                                                                                                                                                                                      

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Textbook Question

Use the given substitution to evaluate the following indefinite integrals. Check your answer by differentiating.                                                                                              

                                                                                                                                                                                        

 βˆ« (6𝓍 + 1) √(3𝓍² + 𝓍) d𝓍 , u = 3𝓍² + 𝓍

Textbook Question

Does a right Riemann sum underestimate or overestimate the area of the region under the graph of a function that is positive and decreasing on an interval [a,b]? Explain.

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Textbook Question

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

                                                                                                                                                                              

 βˆ«β‚Β³ ( 2Λ£ / 2Λ£ + 4 ) d𝓍

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Textbook Question

Symmetry in integrals Use symmetry to evaluate the following integrals.

βˆ«β‚‹Ο€/β‚„^Ο€/⁴ secΒ² x dx

Textbook Question

{Use of Tech} Areas of regions Find the area of the region 𝑅 bounded by the graph of Ζ’ and the 𝓍-axis on the given interval. Graph Ζ’ and show the region 𝑅.                                              

                                                                                                                                                                                    

 Ζ’(𝓍) = 𝓍² (𝓍 ― 2) on [ ―1 , 3]