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What, if anything, does the divergence of 11x2dxand the comparison test tell you about the convergence or divergence of 11x2+1dxand why?

Short Answer

Expert verified

The convergence of11x2dx and comparison test are useful to tell the convergence of 11x2+1dx.

Step by step solution

01

Step 1. Given information.   

We are given two integrals,

11x2dxand

11x2+1dx

02

Step 2. Comparing the integrals. 

Finding the relation between 1x2and 1x2+1in the interval [1,).

x2<x2+1for all x[1,)

1x2>1x2+1for all x[1,)

That is, 1x2+1<1x2for all x[1,)

Apply comparison test for improper integrals to the continuous functions, we get

Since 11x2dxconverges and 1x2+1<1x2in the interval [1,), the improper integral 11x2+1dxmust converge in the interval [1,).

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Most popular questions from this chapter

Complete the square for each quadratic in Exercises 28–33. Then describe the trigonometric substitution that would be appropriate if you were solving an integral that involved that quadratic.

x2-5x+1

For each integral in Exercises 5–8, write down three integrals that will have that form after a substitution of variables.

eudu

Solve the integral:x2+1exdx

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 solving1x24dx.

(b) True or False: The substitution x = 2 sec u is a suitable choice for solving1x24dx.

(c) True or False: The substitution x = 2 tan u is a suitable choice for solving1x2+4dx.

(d) True or False: The substitution x = 2 sin u is a suitable choice for solvingx2+45/2dx

(e) True or False: Trigonometric substitution is a useful strategy for solving any integral that involves an expression of the form x2a2.

(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.

Solvex4+x2dxthe following two ways:

(a) with the substitution u=4+x2;

(b) with the trigonometric substitution x = 2 tan u.

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