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Chapter 5: Q. 34 (page 478)

Use limits of definite integrals to calculate each of the improper integrals in Exercises.

鈭�4鈭�2x-4x2-4x+3dx

Short Answer

Expert verified

The improper integral on [4,鈭�)diverges.

鈭�4鈭�2x-4x2-4x+3dx=鈭�

Step by step solution

01

Step 1. Given information. 

The given integral is the following.

鈭�4鈭�2x-4x2-4x+3dx

02

Step 2. Value of integral. 

Find the integral value.

鈭�4鈭�2x-4x2-4x+3dx=limb鈫�鈭�鈭�4b2x-4x2-4x+3dx=limb鈫�鈭�lnx2-4x+34b=limb鈫�鈭�lnb2-4b+3-ln3=鈭�-ln3=鈭�

So integral on [4,鈭�)diverges

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Most popular questions from this chapter

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

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.

鈭�x4-32+3x2dx

Solve given definite integral.

鈭�04鈥�x3x2+4dx

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.

Solve the integral

鈭�4-xx-2dx
See all solutions

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