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Chapter 5: Q. 6 (page 477)

Why is it very easy to conclude that∫1∞x2dx diverges without making any integration calculations?

Short Answer

Expert verified

The integrand x2is continuous on x=1but not continuous on x=∞. So, the improper integral diverges when integrand is not continuous on the given interval. So, it is easy to conclude that∫1∞x2dxdiverges without making any integration calculations.

Step by step solution

01

Step 1. Given information

∫1∞x2dx.

02

Step 2. The integral over the integrand is infinite.

So, the integral is improper.

The integrand x2is continuous on x=1but not continuous onx=∞.

So, the improper integral diverges when integrand is not continuous on the given interval. So, it is easy to conclude that ∫1∞x2dxdiverges without making any integration calculations.

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Most popular questions from this chapter

Solve given definite integral.

∫12 1x9−x2dx

Describe two ways in which the long-division algorithm for polynomials is similar to the long-division algorithm for integers and then two ways in which the two algorithms are different.

Solve the integral∫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.

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∫1x2−4dx.

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

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

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

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

(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.

Solve∫x4+x2dxthe following two ways:

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(b) with the trigonometric substitution x = 2 tan u.
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