   Chapter 8.7, Problem 65E

Chapter
Section
Textbook Problem

EXPLORING CONCEPTSFinding a Pattern(a) Find f x n ln x   d x for n = 1 ,   2 , and 3. Describe any patterns you notice.(b) Write a general rule for evaluating the integral in part (a) for an integer n ≥ 1 .(c) Verify your rule from part (b) using integration by parts.

(a)

To determine

To calculate: The pattern for values of xnlnxdx for n=1,2,3.

Explanation

Given:

xnlnxdx

Formula used:

Integration by parts for two functions u and v:

u.v=uv(v(u))

The integration formula:

xndx=xn+1n+1

Calculation:

For n=1,

xlnxdx

Solve the above integration by integration by parts,

Therefore,

xlnxdx=lnxxdx(xdx)(lnx)dx=x22lnx(x22)1xdx=x22lnxx24+C

Now for n=2,

x2lnxdx

Solve the above integration by integration by parts,

x2lnxdx=lnxx2dx(x2dx

(b)

To determine
A general rule for evaluating the integral xnlnxdx for integer n1.

(c)

To determine

To prove: The integral:

xnlnxdx=xn+1n+1lnxxn+1(n+1)2

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