   Chapter 8.5, Problem 20E

Chapter
Section
Textbook Problem

# Using Partial Fractions In Exercises 3-20, use partial fractions to find the indefinite integral. ∫ x 16 x 4 − 1 d x

To determine

To calculate: The indefinite integral for the given function, by using partial fractions.

Explanation

Given:

The integral function is x16x41dx.

Formula used:

1xdx=ln|x|+C.

Calculation:

Consider the following indefinite integral,

x16x41dx

The above integral function can be written as,

x16x41=x(4x21)(4x2+1)=x(2x1)(2x+1)(4x2+1)

Now by using partial fraction method we get,

x(2x1)(2x+1)(4x2+1)=A(2x1)+B(2x+1)+Cx+D(4x2+1)x=A(4x2+1)(2x+1)+B(2x1)(4x2+1)+(Cx+D)(2x1)(2x+1)

By simplifying further we get,

At x=12,

12=[A(4(12)2+1)(2(12)+1)+B(2(12)1)(4(12)2+1)+(C(12)+D)(2(12)1)(2(12)+1)]A=18

At x=12,

12=[A(4(12)2+1)(2(12)+1)+B(2(12)1)(4(12)2+1)+(C(12)+D)(2(12)1)(2(12)+1)]B=18

At x=0,

0=A(4(0)2+1)(2(0)+1)+B(2(0)1)(4

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