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Consider the integral -22e53xdx.

(a) Solve this integral by using u-substitution while keeping the limits of integration in terms of x.

(b) Solve the integral again with u-substitution, this time changing the limits of integration to be in terms of u.

Short Answer

Expert verified

(a) The value of integral by using u-substitution while keeping the limits of integration in terms of x is -22e53xdx=-13e1-e11.

(b) The value of integral again with u-substitution, this time changing the limits of integration to be in terms of u is -22e53xdx=-13e-1-e11.

Step by step solution

01

Step 1. Given Information 

Consider the integral -22e53xdx.

(a) Solve this integral by using u-substitution while keeping the limits of integration in terms of x.

(b) Solve the integral again with u-substitution, this time changing the limits of integration to be in terms of u.

02

Part (a) Step 1. Now solving this integral by using u-substitution while keeping the limits of integration in terms of x. 

Let

u=53xdudx=-3-13du=dx
03

Part (a) Step 2. This substitution changes the integral into 

-22e53xdx=-13-22eudu-22e53xdx=-13eu-22-22e53xdx=-13e53x-22-22e53xdx=-13e53×2-e53×(-2)-22e53xdx=-13e56-e5+6-22e53xdx=-13e1-e11

04

Part (b) Step 2. Now solve the integral again with u-substitution, this time changing the limits of integration to be in terms of u.

Let

u=53xdudx=-3-13du=dx

05

Part (b) Step 2. We will now write the limits of integration (x=-2 and x=2) in terms of the new variable u.

When x=-2 we have

u=5-3xu(-2)=5-3(-2)u(-2)=5+6u(-2)=11

When x=2we have

u=5-3xu(2)=5-3(2)u(2)=5-6u(2)=-1

06

Part (a) Step 3. This substitution changes the integral into 

-22e53xdx=-1311-1eudu-22e53xdx=-13eu11-1-22e53xdx=-13e-1-e11

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Most popular questions from this chapter

Problem Zero: Read the section and make your own summary of the material.

Explain why 2xx2+1dxand 1xlnxdxare essentially the same integral after a change of variables.

Which of the integrals that follow would be good candidates for trigonometric substitution? If a trigonometric substitution is a good strategy, name the substitution. If another method is a better strategy, explain that method.

(a)4+x2xdx (b)x4+x2dx

role="math" localid="1648759296940" (c)x24+x2dx (d)16x44+x2dx

Find three integrals in Exercises 39–74 that can be solved by using a trigonometric substitution of the form x=atanu.

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.

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