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Explain why, if x=asecu, then a2sec2ua2is atanuif x<aand is atanuif x>a. Your explanation should include a discussion of domains and absolute values.

Short Answer

Expert verified

Ans:

Theabsolutevaluesresignificantsincetanuispositiveforuin1stquadrantandin2stquadrantitsnegativeforu.

Step by step solution

01

Step 1. Given information:

a2sec2ua2x=asecuatanux<aatanux>a

02

Step 2. Solving the trigonometric substitution:

Suppose x(-,a][a,),u0,π2π2,π, and x=asecu

x2-a2=a2sec2u-a2=a2tan2u=atanuatanu,ifx<aatanu,ifx>aTheabsolutevaluesresignificantsincetanuispositiveforuin1stquadrantandin2stquadrantitsnegativeforu.

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Most popular questions from this chapter

Solve the integral:xexdx

Solve the integral :xex2+1dx

Consider the integral 1x21x2dxfrom the reading at the beginning of the section.

(a) Use the inverse trigonometric substitution u=sin1xto solve this integral.

(b) Use the trigonometric substitution x=sinu to solve the integral.

(c) Compare and contrast the two methods used in parts (a) and (b).

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

u(x)=x2+x+1

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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