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Solve each of the integrals in Exercises 21–70. Some integrals require substitution, and some do not. (Exercise 69 involves a hyperbolic function.)

2xe3x2dx

Short Answer

Expert verified

The solution of the given integral is 2xe3x2dx==13e3x2+C.

Step by step solution

01

Step 1. Given Information 

Solving the given integrals.

2xe3x2dx

02

Step 2. Solving the given integral using substitution method.  

Let

u=3x2dudx=6xdu=6xdx16du=xdx

03

Step 3. This substitution changes the integral into 

2xe3x2dx=26eudu2xe3x2dx=13eu+C2xe3x2dx=13e3x2+C

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Most popular questions from this chapter

Why doesn’t the definite integral231-x2dx make sense? (Hint: Think about domains.)

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)=sec1x

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.

x24x8

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.

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

1udu

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