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Chapter 5: Q. 12 (page 417)

For each function u(x) in Exercises 9鈥�12, write the differential du in terms of the differential dx.

u(x)=1x

Short Answer

Expert verified

The differential du in terms of the differential dx is du=-1x2dx.

Step by step solution

01

Step 1. Given Information

For given function u(x) we have to write the differential du in terms of the differential dx.

u(x)=1x

02

Step 2. Now finding the differential du in terms of the differential dx. 

u(x)=1xu(x)=x-1dudx=-1x-2dudx=-1x2du=-1x2dx

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Most popular questions from this chapter

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

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 solving鈭�1x2鈭�4dx.

(b) True or False: The substitution x = 2 sec u is a suitable choice for solving鈭�1x2鈭�4dx.

(c) True or False: The substitution x = 2 tan u is a suitable choice for solving鈭�1x2+4dx.

(d) True or False: The substitution x = 2 sin u is a suitable choice for solving鈭�x2+4鈭�5/2dx

(e) True or False: Trigonometric substitution is a useful strategy for solving any integral that involves an expression of the form x2鈭�a2.

(f) True or False: Trigonometric substitution doesn鈥檛 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.

Find three integrals in Exercises 27鈥�70 for which either algebra or u-substitution is a better strategy than integration by parts.

Find three integrals in Exercises 27鈥�70 for which a good strategy is to apply integration by parts twice.

Give an example of an integral for which trigonometric substitution is possible but an easier method is available. Then give an example of an integral that we still don鈥檛 know how to solve given the techniques we know at this point.
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