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Why is it very easy to conclude that011exdx converges without making any integration calculations?

Short Answer

Expert verified

The integral 1exis continuous on the interval 0,1. The function 1exhas no vertical asymptotes, so this is not improper integral. So, it is easy to conclude that given integral converges without making any integration calculations.

Step by step solution

01

Step 1. Given information

011exdx.

02

Step 2. The given integral converges.

The integral1exis continuous on the interval0,1.

The function 1exhas no vertical asymptotes, so this is not improper integral.

From the above information, it is easy to conclude that given integral converges without making any integration calculations.

03

Step 3. The graph of y=1ex is shown below:

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Most popular questions from this chapter

Solve the integral: xsinx2dx.

Consider the integral sinxcosxdx.

(a) Solve this integral by using u-substitution with u=sinx and du=cosxdx.

(b) Solve the integral another way, using u-substitution with u=cosx and du=sinxdx.

(c) How must your two answers be related? Use algebra to prove this relationship.

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.

Solve the integral:xlnxdx

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

u2du

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