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Chapter 5: Q. 15 (page 495)

Fill in the blanks to complete each of the following theorem statements:

If the improper integral of fon Idiverges, and 0≤f(x)≤g(x)for all x∈Iwhere the functions are defined, then the improper integral of gon Imust _____.

Short Answer

Expert verified

If the improper integral of fon Idiverges, and 0≤f(x)≤g(x)for all x∈Iwhere the functions are defined, then the improper integral of g on I must diverge.

Step by step solution

01

Step 1. Given information 

If the improper integral of fon Idiverges, and 0≤f(x)≤g(x)for all x∈Iwhere the functions are defined, then the improper integral of g on Imust _____.

02

Step 2. Filling in the blanks to complete the theorem statements 

If the improper integral of fon Idiverges, and 0≤f(x)≤g(x)for all x∈Iwhere the functions are defined, then the improper integral of gon I must diverge.

Improper integrals that are smaller than convergent ones will also converge, and improper integrals that are larger than divergent ones will also diverge. This idea makes intuitive sense because a (positive) quantity that is smaller than a finite number will also be finite and a quantity that is larger than an infinite quantity will also be infinite.

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Most popular questions from this chapter

Why is it okay to use a triangle without thinking about the unit circle when simplifying expressions that result from a trigonometric substitution withx=asinuor x=atanu? Why do we need to think about the unit circle after trigonometric substitution with x=asecu?

Solve the integral:∫lnxdx

Domains and ranges of inverse trigonometric functions: For each function that follows, (a) list the domain and range, (b) sketch a labeled graph, and (c) discuss the domains and ranges in the context of the unit circle.

f(x)=sec−1⁡x

Solve the integral∫x3x2-1dxthree ways:

(a) with the substitution u=x2-1,followed by back substitution;

(b) with integration by parts, choosing localid="1648814744993" u=x2anddv=xx2-1dx;

(c) with the trigonometric substitution x = sec u.

True/False: Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) True or False: The substitution x = 2 sec u is a suitable choice for solving∫1x2−4dx.

(b) True or False: The substitution x = 2 sec u is a suitable choice for solving∫1x2−4dx.

(c) True or False: The substitution x = 2 tan u is a suitable choice for solving∫1x2+4dx.

(d) True or False: The substitution x = 2 sin u is a suitable choice for solving∫x2+4−5/2dx

(e) True or False: Trigonometric substitution is a useful strategy for solving any integral that involves an expression of the form x2−a2.

(f) True or False: Trigonometric substitution doesn’t solve an integral; rather, it helps you rewrite integrals as ones that are easier to solve by other methods.

(g) True or False: When using trigonometric substitution with x=asinu, we must consider the cases x>a and x<-a separately.

(h) True or False: When using trigonometric substitution with x=asecu, we must consider the cases x>a and x<-a separately.
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