   Chapter 7.4, Problem 50E

Chapter
Section
Textbook Problem

# Make a substitution to express the integrand as a rational function and then evaluate the integral. ∫ e x ( e x − 2 ) ( e 2 x + 1 ) d x

To determine

To find: Make a substitution to express the integrand as a rational function and then evaluate the integral ex(ex2)(e2x+1)dx

Explanation

Calculation: Given ex(ex2)(e2x+1)dx

ex(ex2)(e2x+1)dx=ex(ex2)(ex)2+1dx

Let u=ex then du=exdx

ex(ex2)(e2x+1)dx=1(u2)(u2+1)du

Resolving into partial fractions

1(u2)(u2+1)=Au2+Bu+Cu2+1

Then

A(u2+1)+(Bu+C)(u2)(u2)(u2+1)=1(u2)(u2+1)

On comparing the coefficients of u2,u,u0

A=15,B=15,C=25

Thus we have that

1(u2)(u2+1)du=(15(u2)+u25(u+21))du=151u2du+15

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