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Chapter 5: Q. 54 (page 478)

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

鈭�2鈭�1x(x-1)dx

Short Answer

Expert verified

The value isln(2).

Step by step solution

01

Step 1. Given information.

The given integral is鈭�2鈭�1x(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�2鈭�1x(x-1)dx=limB鈫�鈭�鈭�2鈭�1x(x-1)dx=limB鈫�鈭�[ln(|x-1|)-ln(|x|)]2B=limB鈫�鈭�[ln(|B-1|)-ln(|B|)]-[ln(|2-1|)-ln(|2|)]=ln(2)

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Most popular questions from this chapter

Explain why, if x=asinu, then a2鈭�x2=acosu. Your explanation should include a discussion of domains and absolute values.

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Examples: Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading.

(a) An integral with which we could reasonably apply trigonometric substitution with x=tan鈦�u.

(b) An integral with which we could reasonably apply trigonometric substitution with x=4sec鈦�u.

(c) An integral with which we could reasonably apply trigonometric substitution with x鈭�2=3sin鈦�u.

Explain why it makes sense to try the trigonometric substitution x=sec鈦�uif an integrand involves the expression x2鈭�1
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