   Chapter 7.4, Problem 53E

Chapter
Section
Textbook Problem

# Use integration by parts, together with the techniques of this section, to evaluate the integral. ∫ ln ( x 2 − x + 2 ) d x

To determine

To Find: Use integration by parts, together with the techniques of this section, to evaluate the integral ln(x2x+2)dx

Explanation

Calculation: Given ln(x2x+2)dx

Integration parts, we get

ln(x2x+2)dx=ln(x2x+2)dx[ddx(ln(x2x+2))dx ]dx=xln(x2x+2)(2x1)xx2x+2dx=xln(x2x+2)2x2xx2x+2dx

Resolving into partial fractions 2x2xx2x+2dx

2x2xx2x+2dx=2+x4x2x+2dx=2+x4(x12)2+74dx=[2+x12(x12)2+7472(x12)2+74]dx

Substitute x12=u then dx=du

=2+uu2+7472u2+74du

=2du+122uu2+74du72duu2+74

=2u+12ln(u2+74)72

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