   Chapter 6.1, Problem 53E ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

#### Solutions

Chapter
Section ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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# Verifying Formulas In Exercises 53 and 54, use integration by parts to verify the formula. ∫ x n ln   x   d x = x n + 1 ( n + 1 ) 2 [ − 1 + ( n + 1 ) ln x ] + C ,   n ≠ − 1

To determine

To prove: The indefinite integral xnlnxdx=xn+1(n+1)2[1+(n+1)lnx]+C,n1.

Explanation

Given Information:

The provide indefinite integral is xnlnxdx.

Formula used:

xndx=xn+1n+1,n1,d(lnx)dx=1x

Proof:

Consider the indefinite integral xnlnxdx

Here,

dv=xndx and u=lnx

First find v,

dv=xndxdv=xndx

On further solving,

v=xn+1n+1 …...…... (1)

Now find du,

u=lnx

Differentiate both side with respect x,

dudx=d(lnx)dxdudx=1x

And,

du=1xdx …...…..

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