   Chapter 7.8, Problem 56E

Chapter
Section
Textbook Problem

# Evaluate ∫ 2 ∞ 1 x x 2 − 4   d x by the same method as in Exercise 55.

To determine

To evaluate:

The integral of the expression 21xx24dx.

Explanation

Given:

21xx24dx

Formulae used:

The integration by substitution

Consider 21xx24dx

Note that the integral is improper on two counts, since the function is not defined at x=2 and the upper limit is not finite. Thus we shall first split it into two improper integrals as follows.

21xx24dx=lima2+a31xx24dx+lima3a1xx24dx

We shall use the following trigonometric substitution to solve both integrals.

2sect=xt=sec1x22secttantdt=dx

First we shall evaluate the indefinite integral and apply the limit afterwards.

1xx24dx=2secttantdt2sect4sec2t4=tantdt4(sec2t1)=tantdt4tan2t=12dt=12t

Re-substituting for t we have

1xx24dx=12sec1x2

Thus

21xx24dx=lima2+a31xx24dx+lima3a1xx24dx=lima2+[12

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