   Chapter 7.8, Problem 17E

Chapter
Section
Textbook Problem

# Determine whether each integral is convergent or divergent. Evaluate those that are convergent. ∫ 1 ∞ 1 x 2 + x   d x

To determine

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

Explanation

Given:

11x2+xdx.

Formula used:

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

11x2+xdx

This is an improper integral so we replace  with t and evaluate the limit of definite integral as limt.

So,11x2+xdx=limt1t1x2+xdx (I)

Taking only integral part of equation (I),

1t1x2+xdx

1x2+x=Ax+B(x+1) [By partial fraction]

1=A(x+1)+B(x)1=Ax+A+Bx0x+1=(A+B)x+A

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

A=1

And

A+B=0

So,

B=1

So,1x2+x=1x+1x+1

1t1x2+xdx=1t(1x1x+1)dx

=1t1xdx1t1x+1

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