   Chapter 7.5, Problem 77E

Chapter
Section
Textbook Problem

# Evaluate the integral. ∫ x e x 1 + e x d x

To determine

To find:

xexdx1+ex

Explanation

Formula used: 1x2-a2=ln|x-ax+a|

Calculations: In the given integral

xexdx1+ex

Substitute ex=u,dx=exdu, and  x=ln u

Use integration by parts zdy=yz-ydz with z=lnu, dy=1+u

dz=1u du and y=21+u

Thus

I=lnu du1+u=21+u lnu-21+uudu

Now we shall try to solves

I2=1+uudu

Let 1+u=v, that is v2-1=u,  so du=2vdv=21+u dv

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