   Chapter 7.8, Problem 59E

Chapter
Section
Textbook Problem

# Find the values of p for which the integral converges and evaluate the integral for those values of p. ∫ 0 1 x p ln x     d x

To determine

To find:

The value of p for which the given integral converges.

Explanation

Given:

01xplnxdx

Formulae used:

The integration by parts

0xf(x)g(x)dx=f(x)g(x)dx(df(x)dxg(x)dx)dx

Consider 01xplnxdx

Hence, use the formulae of integration by parts

0xf(x)g(x)dx=f(x)g(x)dx(df(x)dxg(x)dx)dx

Now according to the given expression, the value of the given expression 01xplnxdx is

01xplnxdx=[lnx1p+1xp+1]01011p+1xp+11xdx=ln11p+1(1)p+1limx0lnx1p+1xp+11p+101xpdx=1p+1limx0lnxxp+11(p+1)2

If p<1 then,

01xplnxdx</

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