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Chapter 5: Q. 25 (page 464)

In Exercises 20鈥�27, use reference triangles and the unit circle to write the given trigonometric compositions as algebraic functions.

  • If x鈭�5=3sinu, then write tan2uas an algebraic function of x.

Short Answer

Expert verified

Ans:tan2u=(x-5)29-(x-5)2

Step by step solution

01

Step 1. Given information:

x鈭�5=3sinu

02

Step 2. Finding the integrals using trigonometric substitution:

The expression is as follows.

x鈭�5=3sinu

The objective is to write tan2uas an algebraic function of x

The algebraic function is as follows.

tan2u=sin2ucos2u=x2(a2-x2)2=(x-5)2(9-(x-5)2)2=(x-5)29-(x-5)2

Therefore, the value is(x-5)29-(x-5)2

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Most popular questions from this chapter

Solve each of the integrals in Exercises 39鈥�74. Some integrals require trigonometric substitution, and some do not. Write your answers as algebraic functions whenever possible.

鈭�9-x2xdx

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?

Solve given definite integral.

鈭�1/41/2鈥�1x21鈭�x2dx

Solve the integral鈭�x3x2-1dxthree ways:

(a) with the substitution u=x2-1,followed by back substitution;

(b) with integration by parts, choosing localid="1648814744993" u=x2anddv=xx2-1dx;

(c) with the trigonometric substitution x = sec u.

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)=sec鈭�1鈦�x
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