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Consider the integral x(x21)2dx.

(a) Solve this integral by using u-substitution.

(b) Solve the integral another way, using algebra to multiply out the integrand first.

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

Short Answer

Expert verified

(a) The value of integral by using u-substitutionx(x21)2dx=(x21)36+C

(b) The the value integral using algebra to multiply out the integrand first is x(x21)2dx=x66-x42+x22+C.

(c) The value of both answer differ by a constant(x21)36=x6-3x4+3x26-16.

Step by step solution

01

Step 1. Given Information 

Consider the integral x(x21)2dx.

(a) Solve this integral by using u-substitution.

(b) Solve the integral another way, using algebra to multiply out the integrand first.

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

02

Part (a) Step 1. Solve this integral by using u-substitution.

Let u=x21

dudx=2xdu=2xdx12du=xdx

03

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

x(x21)2dx=12u2dux(x21)2dx=12u2+12+1+Cx(x21)2dx=12u33+Cx(x21)2dx=u36+Cx(x21)2dx=(x21)36+C

04

Part (b) Step 1. Solving integral using algebra to multiply out the integrand first 

x(x21)2dx=x(x2)22×1×x2+(1)2dxx(x21)2dx=xx42x2+1dxx(x21)2dx=(x·x42·x·x2+x·1)dxx(x21)2dx=(x52x3+x)dx

05

Part (b) Step 2. The integral after simplifying ∫x(x2−1)2dx=∫(x5−2x3+x)dx 

Now solving.

x(x21)2dx=x5dx2x3dx+xdxx(x21)2dx=x5+15+12x3+13+1+x1+11+1x(x21)2dx=x66-2·x44+x22+Cx(x21)2dx=x66-x42+x22+C

06

Part (c) Step 1. Using algebra to prove this relationship. 

The value of integral in part (a) is

x(x21)2dx=(x21)36

The value of integral in part (b) is

x(x21)2dx=x6-3x4+3x26

The value of both answer differ by a constant

(x21)36=x6-3x4+3x26-16

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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?

Find three integrals in Exercises 21–70 in which the denominator of the integrand is a good choice for a substitution u(x).

Solve given integrals by using polynomial long division to rewrite the integrand. This is one way that you can sometimes avoid using trigonometric substitution; moreover, sometimes it works when trigonometric substitution does not apply.

x4-32+3x2dx

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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