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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 5.5.95a
Chapter 5, Problem 5.5.95a

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume ƒ, ƒ', and ƒ'' are continuous functions for all real numbers.                                                                                                                                                           
                                                                                                                                                                    
(a) ∫ ƒ(𝓍) ƒ'(𝓍) d𝓍 = ½ (ƒ(𝓍))² + C.

Verified step by step guidance
1
Step 1: Recall the integration by substitution method. To evaluate ∫ ƒ(𝓍) ƒ'(𝓍) d𝓍, consider substituting u = ƒ(𝓍). This substitution simplifies the integral.
Step 2: Compute the derivative of u with respect to 𝓍. Since u = ƒ(𝓍), then du/d𝓍 = ƒ'(𝓍), or equivalently, du = ƒ'(𝓍) d𝓍.
Step 3: Rewrite the integral in terms of u. Substituting u = ƒ(𝓍) and du = ƒ'(𝓍) d𝓍, the integral becomes ∫ u du.
Step 4: Solve the integral ∫ u du. The antiderivative of u with respect to u is (1/2)u² + C, where C is the constant of integration.
Step 5: Substitute back u = ƒ(𝓍) into the result. This gives (1/2)(ƒ(𝓍))² + C, which matches the given statement. Therefore, the statement is true.

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Key Concepts

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

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus links the concept of differentiation and integration, stating that if a function is continuous on an interval, then the integral of its derivative over that interval equals the change in the function's values. This theorem is crucial for evaluating definite integrals and understanding the relationship between a function and its antiderivative.
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Fundamental Theorem of Calculus Part 1

Integration by Substitution

Integration by substitution is a technique used to simplify the process of integration by changing the variable of integration. It involves substituting a part of the integrand with a new variable, which can make the integral easier to solve. This method is particularly useful when dealing with composite functions.
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Continuous Functions

A function is continuous if it does not have any breaks, jumps, or holes in its graph. For the statements in the question, the continuity of the functions ƒ, ƒ', and ƒ'' ensures that the properties of limits and integrals apply, allowing for the application of the Fundamental Theorem of Calculus and ensuring that the results derived from these functions are valid.
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Related Practice
Textbook Question

Zero net area Consider the function ƒ(𝓍) = 𝓍² ― 4𝓍 .

(a) Graph ƒ on the interval 𝓍 ≥ 0.

Textbook Question

Average value with a parameter Consider the function ƒ(𝓍) = a𝓍 (1―𝓍) on the interval [0, 1], where a is a positive real number.

(a) Find the average value of ƒ as a function of a .

Textbook Question

Mass from density A thin 10-cm rod is made of an alloy whose density varies along its length according to the function shown in the figure. Assume density is measured in units of g/cm. In Chapter 6, we show that the mass of the rod is the area under the density curve.

(a) Find the mass of the left half of the rod (0 ≤ x ≤ 5) .

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Textbook Question

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

(a) Consider the linear function ƒ(𝓍) = 2x + 5 and the region bounded by its graph and the x-axis on the interval [3,6]. Suppose the area of this region is approximated using midpoint Riemann sums. Then the approximations give the exact area of the region for any number of subintervals.

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Textbook Question

Area functions for constant functions Consider the following functions ƒ and real numbers a (see figure).

(a) Find and graph the area function A(𝓍) = ∫ₐˣ ƒ(t) dt for ƒ.

ƒ(t) = 5 , a = 0

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Textbook Question

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

(a) If ƒ is a constant function on the interval [a,b], then the right and left Riemann sums give the exact value of ∫ₐᵇ ƒ(𝓍) d𝓍, for any positive integer n.