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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 5.5.15f
Capitolo 5, Problema 5.5.15f

Use Table 5.6 to evaluate the following indefinite integrals.                                                                                                               
                                                                                                                                                                  
 (f) ∫ d𝓍/√36 ―𝓍²

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Step 1: Recognize the integral ∫ d𝓍/√36 ― 𝓍² as a standard form from Table 5.6. This corresponds to the formula for the arcsine function: ∫ dx/√a² - x² = arcsin(x/a) + C, where 'a' is a constant.
Step 2: Identify the value of 'a' in the given integral. Here, the term √36 indicates that a² = 36, so a = 6.
Step 3: Rewrite the integral in the standard form by substituting 'a' into the formula. The integral becomes ∫ d𝓍/√6² - 𝓍².
Step 4: Apply the formula for the arcsine function. Using the standard result, the integral evaluates to arcsin(𝓍/6) + C, where C is the constant of integration.
Step 5: Conclude the solution by stating that the indefinite integral has been evaluated using the arcsine formula, and the result is expressed in terms of arcsin(𝓍/6) + C.

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Indefinite Integrals

Indefinite integrals represent a family of functions whose derivative is the integrand. They are expressed without limits of integration and include a constant of integration, typically denoted as 'C'. The process of finding an indefinite integral is often referred to as antiderivation, where one seeks a function whose derivative matches the given function.
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Introduction to Indefinite Integrals

Integration Techniques

Various techniques exist for evaluating integrals, including substitution, integration by parts, and using integral tables. In this context, Table 5.6 likely contains standard integrals that can be directly applied to simplify the evaluation process. Recognizing which technique or table entry to use is crucial for efficiently solving integrals.
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Integration by Parts for Definite Integrals

Trigonometric Substitution

Trigonometric substitution is a method used to simplify integrals involving square roots of quadratic expressions. By substituting a variable with a trigonometric function, such as x = a sin(θ) or x = a tan(θ), the integral can often be transformed into a more manageable form. This technique is particularly useful for integrals like ∫ dx/√(a² - x²), which can be evaluated using the properties of trigonometric functions.
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Use Table 5.6 to evaluate the following indefinite integrals.                                                                                                               

                                                                                                                                                                  

 (e) ∫ d𝓍/(81 + 9𝓍²) (Hint: Factor a 9 out of the denominator first.)  

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Sigma notation Evaluate the following expressions.                                                                                                                                          

(e)     3                                                                                                                                                                               

       ∑  (2m + 2) / 3                                                                                                                                                                          

      m =1                         

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(d) F(4)

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Area functions The graph of ƒ is shown in the figure. Let A(x) = ∫₀ˣ ƒ(t) dt and F(x) = ∫₂ˣ ƒ(t) dt be two area functions for ƒ. Evaluate the following area functions.

(g) F(2)

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Sigma notation Evaluate the following expressions.                                                                                                                                          

  (f)      3                                                                                                                                                                               

       ∑ (3j ― 4)                                                                                                                                                                          

      j =1                         

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Properties of integrals Consider two functions ƒ and g on [1,6] such that ∫₁⁶ƒ(𝓍) d𝓍 = 10 and ∫₁⁶g(𝓍) d𝓍 = 5, ∫₄⁶ƒ(𝓍) d𝓍 = 5 , and ∫₁⁴g(𝓍) d𝓍 = 2. Evaluate the following integrals.


(f) ∫₄¹ 2f(𝓍) d𝓍

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