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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 5.5.15e
Chapter 5, Problem 5.5.15e

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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Step 1: Start by factoring a 9 out of the denominator. Rewrite the integral as ∫ d𝓍 / (9(9 + 𝓍²)). This simplifies the denominator and prepares it for further manipulation.
Step 2: Factor out the constant 1/9 from the integral. This gives (1/9) ∫ d𝓍 / (9 + 𝓍²). Constants can be factored out of integrals to simplify calculations.
Step 3: Recognize the standard form of the integral. The denominator (9 + 𝓍²) matches the form a² + 𝓍², where a = 3. This suggests using the formula ∫ d𝓍 / (a² + 𝓍²) = (1/a) arctan(𝓍/a) + C.
Step 4: Apply the formula. Substitute a = 3 into the formula, resulting in (1/9) * (1/3) arctan(𝓍/3) + C. Combine the constants to simplify the expression further.
Step 5: Write the final simplified integral expression as (1/27) arctan(𝓍/3) + C. This is the indefinite integral of the given function.

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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 followed by the function and the differential, and they include a constant of integration, C. Understanding how to evaluate indefinite integrals is crucial for solving problems in calculus, as they provide the antiderivative of a function.
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Introduction to Indefinite Integrals

Factoring in Integration

Factoring is a technique used to simplify expressions, making them easier to integrate. In the context of the given problem, factoring a common term from the denominator can transform the integral into a more manageable form. This step is essential for applying integration techniques, such as substitution or recognizing standard integral forms.
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Standard Integral Forms

Standard integral forms are well-known results that provide the antiderivatives of specific functions. These forms are often found in integral tables and can significantly expedite the integration process. Recognizing when an integral matches a standard form allows for quick evaluation, which is particularly useful in solving complex integrals efficiently.
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Related Practice
Textbook Question

Left and right Riemann sums Complete the following steps for the given function, interval, and value of n.

f(x) = x + 1 on [0,4]; n = 4

(d) Calculate the left and right Riemann sums.                                                                                                                                                

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Textbook Question

Sigma notation Evaluate the following expressions.                                                                                                                                          

(e)     3                                                                                                                                                                               

       ∑  (2m + 2) / 3                                                                                                                                                                          

      m =1                         

Textbook Question

{Use of Tech} Approximating definite integrals Complete the following steps for the given integral and the given value of n. 

(d) Determine which Riemann sum (left or right) underestimates the value of the definite integral and which overestimates the value of the definite integral.


∫₀^π/2 cos 𝓍 d𝓍 ; n = 4

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Textbook Question

Area functions The graph of ƒ is shown in the figure. Let A(x) = ∫₋₂ˣ ƒ(t) dt and F(x) = ∫₄ˣ ƒ(t) dt be two area functions for ƒ. Evaluate the following area functions.

(d) F(4)

Textbook Question

Sigma notation Evaluate the following expressions.                                                                                                                                          

  (f)      3                                                                                                                                                                               

       ∑ (3j ― 4)                                                                                                                                                                          

      j =1                         

Textbook Question

Use Table 5.6 to evaluate the following indefinite integrals.                                                                                                               

                                                                                                                                                                  

 (f) ∫ d𝓍/√36 ―𝓍²