   Chapter 8.2, Problem 69E

Chapter
Section
Textbook Problem

# Proof In Exercises 71-76, use integration by parts to prove the formula. (For Exercises 71-74, assume that n is a positive integer.) ∫ x n ln x   d x = x n + 1 ( n + 1 ) 2 [ − 1 + ( n + 1 ) ln x ] + C

To determine

To prove: The formula given as, xnlnxdx=xn+1(n+1)2[1+(n+1)lnx]+C.

Explanation

Given:

The formula is:

xnlnxdx=xn+1(n+1)2[1+(n+1)lnx]+C

Formula used:

Integration by parts formula, udv=uvvdu

Proof:

Consider the integral, xnlnxdx.

Let u=lnx.

Now, differentiate both the sides with respect to x.

Then,

ddxu=ddxlnxdudx=1xdu=1xdx

And let, dv=xndx.

Now, integrate both the sides.

Then,

dv=xndxv=xn+1(n+1)

Now, apply integration by parts:

udv=uvvdu

Substitute the values

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