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

1x+1x2dx

Short Answer

Expert verified

The improper integral 1x+1x2dxdiverges.

Step by step solution

01

Step 1. Given information. 

The given integral is the following.

1x+1x2dx

02

Step 2. Value of integral. 

Consider functionf(x)=x+1x2

localid="1648879879677" f(x)=x+1x2=xx2+1x2=1x+1x21x+1x2>1x2

In the interval [1,)we have localid="1648879786507" x+1x2>1x2.

As we know that localid="1648884171467" 11x2dxis converse and integral of functions less than 1x2will be converse.

But x+1x21x2so it cannot be converse.

so1x+1x2is diverse.

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Most popular questions from this chapter

Explain why 2xx2+1dxand 1xlnxdxare essentially the same integral after a change of variables.

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

x24x8

True/False: Determinewhethereachofthestatementsthat follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) True or False: f(x)=x+1x-1is a proper rational function.

(b) True or False: Every improper rational function can be expressed as the sum of a polynomial and a proper rational function.

(c) True or False: After polynomial long division of p(x) by q(x), the remainder r(x) has a degree strictly less than the degree of q(x).

(d) True or False: Polynomial long division can be used to divide two polynomials of the same degree.

(e) True or False: If a rational function is improper, then polynomial long division must be applied before using the method of partial fractions.

(f) True or False: The partial-fraction decomposition of x2+1x2(x-3)is of the form Ax2+Bx-3

(g) True or False: The partial-fraction decomposition of x2+1x2(x-3)is of the form Bx+Cx2+Ax-3.

(h) True or False: Every quadratic function can be written in the formA(x-k)2+C

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

Find three integrals in Exercises 21–70 in which the denominator of the integrand is a good choice for a substitution u(x).

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