   Chapter 7.8, Problem 18E

Chapter
Section
Textbook Problem

# Determine whether each integral is convergent or divergent. Evaluate those that are convergent. ∫ 2 ∞ d v v 2 + 2 v − 3

To determine

whether the given integral is convergent or divergent, evaluate it if convergent.

Explanation

Given:

21v2+2v3dv

Formula used:

1xdx=logx+clogxlogy=logxylogx+logy=logxy

21v2+2v3dv

This is an improper integral so we replace  as t and evaluate as limt so,

21v2+2v3dv=limt2t1v2+2v3dv (I)

Taking only integral part of equation (I),

2t1v2+2v3dv=2t1v2+3vv3dv=2t1v(v+3)1(v+3)dv

=2t1(v1)(v+3)dv (II)

1(v+3)(v1)=A(v+3)+B(v1) [By partial fraction]

1=A(v1)+B(v+3)1=AvA+Bv+3B1=(A+B)vA+3B

By comparison of L.H.S and R.H

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