Chapter 8.7, Problem 65E

### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347

Chapter
Section

### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347
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:

âˆ«xnlnxdx

Formula used:

Integration by parts for two functions u and v:

âˆ«u.v=uâ‹…âˆ«vâˆ’âˆ«(âˆ«v(uâ€²))

The integration formula:

âˆ«xndx=xn+1n+1

Calculation:

For n=1,

âˆ«xlnxdx

Solve the above integration by integration by parts,

Therefore,

âˆ«xlnxdx=lnxâˆ«xdxâˆ’âˆ«(âˆ«xdx)â‹…(lnx)dx=x22lnxâˆ’âˆ«(x22)1xdx=x22lnxâˆ’x24+C

Now for n=2,

âˆ«x2lnxdx

Solve the above integration by integration by parts,

âˆ«x2lnxdx=lnxâˆ«x2dxâˆ’âˆ«(âˆ«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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