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Chapter 5: Q. 52 (page 496)

The average value of the function f(x)=2x-1(x+2)(x2+1) on the interval0,10.

Short Answer

Expert verified

The average value is0.052.

Step by step solution

01

Given

f(x)=2x-1(x+2)(x2+1),0,10

02

Calculate the average value.

The average value of a function in the interval is given as,

Average=1b-a鈭�abf(x)dx=110鈭�0102x-1(x+2)(x2+1)dx=110鈭�010-1x+2+xx2+1Byusingpartialfractionmethod=110-lnx+2+12lnx2+1010=110-ln12+12ln101--ln2=11012ln101+ln2-ln12=110脳0.52=0.052

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Most popular questions from this chapter

Complete the square for each quadratic in Exercises 28鈥�33. Then describe the trigonometric substitution that would be appropriate if you were solving an integral that involved that quadratic.

2x2鈭�4x+1

Solve the integral

鈭�1x2x2-9dx

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

鈭�eudu

Problem Zero: Read the section and make your own summary of the material.

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.
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