   Chapter 8.5, Problem 31E

Chapter
Section
Textbook Problem

# Finding an Indefinite Integral In Exercises 25-32, use substitution and partial fraction to find the indefinite integral. ∫ e x ( e x − 1 ) ( e x + 4 ) d x .

To determine

To calculate: The solution of the given indefinite integral using partial fractions.

Explanation

Given: The indefinite integral ex(ex1)(ex+4)dx.

Calculation:

Consider the following indefinite integral

ex(ex1)(ex+4)dx.

Let, u=exdu=exdx

Substituting the integral,

ex(ex1)(ex+4)dx=ex(u1)(u+4)

Now, decomposition into partial fractions,

1(u1)(u+4)=A(u1)+B(u+4)

Simplify by multiplyingwith the common denominator

1=A(u+4)+B(u1)if u=1 then1=A(1+4)+B(11)

Hence,

A=15if u=4 then1=A(4+4)+B(41)B=15B=15

Thus,

1(u1)(u+4)=Au1

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