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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 5.5.15d
Chapter 5, Problem 5.5.15d

Use Table 5.6 to evaluate the following indefinite integrals.                                                                                                               
                                                                                                                                                                  
 (d) ∫ cos 𝓍/7 d𝓍

Verified step by step guidance
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Step 1: Recognize that the integral involves a cosine function divided by a constant. The general formula for the integral of cos(𝓍) is ∫cos(𝓍)d𝓍 = sin(𝓍) + C, where C is the constant of integration.
Step 2: Notice that the argument of the cosine function is scaled by a constant factor (1/7). To handle this, use the substitution rule for integrals. Let u = 𝓍/7, which implies du = (1/7)d𝓍.
Step 3: Rewrite the integral in terms of u. Substituting u = 𝓍/7 and du = (1/7)d𝓍, the integral becomes ∫cos(u) * 7 du.
Step 4: Apply the formula for the integral of cos(u). The integral of cos(u) is sin(u), so the integral becomes 7 * sin(u) + C.
Step 5: Substitute back u = 𝓍/7 to express the result in terms of the original variable. The final expression is 7 * sin(𝓍/7) + C.

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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) since the derivative of a constant is zero. Understanding how to evaluate indefinite integrals is crucial for solving problems in calculus, as they are foundational for determining areas under curves and solving differential equations.
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Introduction to Indefinite Integrals

Integration Techniques

Integration techniques are methods used to evaluate integrals that may not be straightforward. Common techniques include substitution, integration by parts, and using integral tables. In this context, referring to Table 5.6 suggests that specific integrals have been pre-calculated, allowing students to apply these results directly to evaluate the given integral without performing the entire calculation from scratch.
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Integration by Parts for Definite Integrals

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are fundamental in calculus and are often encountered in integration problems. The integral of cosine, for example, is a common form that can be evaluated using known results or integration techniques. Understanding the properties and graphs of these functions is essential for recognizing patterns and applying the appropriate integration methods effectively.
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Introduction to Trigonometric Functions
Related Practice
Textbook Question

Displacement from a velocity graph Consider the velocity function for an object moving along a line (see figure).

(d) Assuming the velocity remains 10 m/s, for t β‰₯ 5, find the function that gives the displacement between t = 0 and any time t β‰₯ 5.

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Textbook Question

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

{Use of Tech} Ζ’(𝓍) = e Λ£/β‚‚ on [1,4]; n = 6

(d) Calculate the left and right Riemann sums. 

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.


βˆ«β‚β· 1/𝓍 d𝓍 ; n = 6

Textbook Question

Properties of integrals Use only the fact that βˆ«β‚€β΄ 3𝓍 (4 ―𝓍) d𝓍 = 32, and the definitions and properties of integrals, to evaluate the following integrals, if possible.

(d) βˆ«β‚€βΈ 3𝓍(4 ― 𝓍) d(𝓍)

Textbook Question

Properties of integrals Suppose βˆ«β‚€Β³Ζ’(𝓍) d𝓍 = 2 , βˆ«β‚ƒβΆΖ’(𝓍) d𝓍 = ―5 , and βˆ«β‚ƒβΆg(𝓍) d𝓍 = 1. Evaluate the following integrals.

(d) βˆ«β‚†Β³ (Ζ’(𝓍) + 2g(𝓍)) d𝓍

Textbook Question

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

                                                                                                                                                                                     (d) If A(𝓍) = 3𝓍²― 𝓍― 3 is an area function for Ζ’, then                                                                                                                                   

     B(𝓍) = 3𝓍² ― 𝓍 is also an area function for Ζ’.