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Chapter 5: Q. 53 (page 478)

Use limits of definite integrals to calculate each of the improper integrals in Exercises 21鈥�56.

鈭�121x(x-1)dx

Short Answer

Expert verified

The answer is鈭�.

Step by step solution

01

Step 1. Given information.

The given integral is鈭�121x(x-1)dx.

02

Step 2. Value of the integral.

On solving,

鈭�1x(x-1)dx=鈭�1x-1-1xdx=ln(|x-1|)-ln(|x|)Now,鈭�121x(x-1)dx=limA鈫�1鈭�A21x(x-1)dx=limA鈫�1[ln(|x-1|)-ln(|x|)]A2=[ln(|2-1|)-ln(|2|)]-limA鈫�1[ln(|A-1|)-ln(|A|)]=鈭�

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Most popular questions from this chapter

Explain how to know when to use the trigonometric substitutions x=asin鈦�u,x=atan鈦�u,andx=asec鈦�u, Describe the trigonometric identity and the triangle that will be needed in each case. What are the possible values for xand uin each case?

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

Solve the integral:鈭�3x+1xdx

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 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�x3-3x2+2x-3x2+1dx
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