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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 5.5.49
Chapter 5, Problem 5.5.49

Definite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals.                                                                                                                         
                                                                                                                                                                              
 ∫₁³ ( 2ˣ / 2ˣ + 4 ) d𝓍

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Step 1: Observe the integral ∫₁³ (2ˣ / (2ˣ + 4)) d𝓍. To simplify this, consider using a substitution method. Let u = 2ˣ + 4, which simplifies the denominator.
Step 2: Differentiate u with respect to x. Since u = 2ˣ + 4, du/dx = ln(2) * 2ˣ. Rearrange to express dx in terms of du: dx = du / (ln(2) * 2ˣ).
Step 3: Substitute u and dx into the integral. Replace 2ˣ in the numerator and denominator using the substitution u = 2ˣ + 4. The integral becomes ∫ (1 / u) * (du / ln(2)).
Step 4: Factor out constants from the integral. The constant 1/ln(2) can be factored out, leaving (1/ln(2)) ∫ (1/u) du.
Step 5: Evaluate the integral of 1/u with respect to u. The result is ln|u|. Substitute back u = 2ˣ + 4 and apply the limits of integration (x = 1 to x = 3) to find 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. 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

Change of Variables

Change of variables, or substitution, is a technique used in integration to simplify the integrand. By substituting a new variable for an existing one, the integral can often be transformed into a more manageable form. This method is particularly useful when dealing with complex functions or when the limits of integration need to be adjusted accordingly.
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Integration Techniques

Integration techniques encompass various methods used to evaluate integrals, including substitution, integration by parts, and using tables of integrals. These techniques help in solving integrals that may not be straightforward. Familiarity with these methods, such as those found in Table 5.6, allows for efficient evaluation of definite integrals in calculus.
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Related Practice
Textbook Question

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

                                                                                                                                                                    

 ∫ [(√𝓍 + 1)⁴ / 2√𝓍 d𝓍

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

A midpoint Riemann sum Approximate the area of the region bounded by the graph of ƒ(𝓍) = 100 ― x² and the x-axis on [0, 10] with n = 5 subintervals. Use the midpoint of each subinterval to determine the height of each rectangle (see figure).

Textbook Question

Symmetry in integrals Use symmetry to evaluate the following integrals.

∫₋π/₄^π/⁴ sec² x 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.                                                                                  

                                                                                                                                                                    

 ∫ d𝓍 / (√1 ― 9𝓍²)

Textbook Question

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

                                                                                                                                                                                    

 ƒ(𝓍) = 𝓍² (𝓍 ― 2) on [ ―1 , 3]