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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 5.5.16c
Chapter 5, Problem 5.5.16c

Use Table 5.6 to evaluate the following definite integrals.                                                                                                                    
 (c) ∫₃√₂^⁶ d𝓍/(𝓍² ―9)

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Step 1: Recognize that the integral involves a rational function of the form 1/(x² - a²). This suggests that the integral can be evaluated using a standard formula from Table 5.6: ∫ dx/(x² - a²) = (1/(2a)) ln| (x - a)/(x + a) | + C.
Step 2: Identify the value of 'a' in the denominator x² - 9. Here, x² - 9 can be rewritten as x² - 3², so a = 3.
Step 3: Apply the formula for the definite integral. Since the integral is definite, the limits of integration are given as √2 (lower limit) and 6 (upper limit). The formula becomes: ∫₃√₂⁶ dx/(x² - 9) = (1/(2 * 3)) [ln| (x - 3)/(x + 3) |] evaluated from x = √2 to x = 6.
Step 4: Simplify the constant factor outside the logarithm. The constant factor (1/(2 * 3)) simplifies to 1/6. So the integral becomes: (1/6) [ln| (x - 3)/(x + 3) |] evaluated from x = √2 to x = 6.
Step 5: Substitute the limits of integration into the logarithmic expression. First, substitute x = 6 into ln| (x - 3)/(x + 3) |, then substitute x = √2 into the same expression. Finally, subtract the result at x = √2 from the result at x = 6 to compute the definite integral.

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Key Concepts

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

Definite Integrals

A definite integral represents the signed area under a curve between two specified limits on the x-axis. It is denoted as ∫_a^b f(x) dx, where 'a' and 'b' are the lower and upper limits, respectively. The result of a definite integral is a numerical value that quantifies the accumulation of the function's values over the interval [a, b].
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Definition of the Definite Integral

Integration Techniques

Integration techniques are methods used to evaluate integrals that may not be solvable by basic antiderivatives. Common techniques include substitution, integration by parts, and partial fraction decomposition. Understanding these techniques is essential for simplifying complex integrals into manageable forms, allowing for easier evaluation.
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Table of Integrals

A table of integrals is a reference tool that lists common integrals and their corresponding antiderivatives. It provides quick access to solutions for frequently encountered functions, saving time and effort in calculations. Familiarity with such tables can significantly aid in evaluating definite integrals efficiently, especially when dealing with standard forms.
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Related Practice
Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

(c) The average value of a linear function on an interval [a, b] is the function value at the midpoint of [a, b] .

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.


∫₁⁷ 1/𝓍 d𝓍 ; n = 6

Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

(c) For an increasing or decreasing nonconstant function on an interval [a,b] and a given value of n, the value of the midpoint Riemann sum always lies between the values of the left and right Riemann sums.

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Textbook Question

Properties of integrals Suppose ∫₀³ƒ(𝓍) d𝓍 = 2 , ∫₃⁶ƒ(𝓍) d𝓍 = ―5 , and ∫₃⁶g(𝓍) d𝓍 = 1. Evaluate the following integrals.

(d) ∫₆³ (ƒ(𝓍) + 2g(𝓍)) d𝓍

Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.                                                                          

                                                                                                                                                                                     (c) The functions p(𝓍) = sin 3𝓍 and q(𝓍) = 4 sin 3𝓍 are antiderivatives of the same function. 

Textbook Question

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

(c) Calculate the left and right Riemann sums for the given value of n.

∫₁⁷ 1/𝓍 d𝓍 ; n = 6