   Chapter 8.5, Problem 18E

Chapter
Section
Textbook Problem

# Using Partial Fractions In Exercises 3-20, use partial fractions to find the indefinite integral. ∫ 6 x x 3 − 8 d x

To determine

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

Explanation

Given:

The integral function is 6xx38dx.

Formula used:

1xdx=ln|x|+C.

Calculation:

Consider the following indefinite integral,

6xx38dx

The above integral function can be written as,

6xx38=6xx323=6x(x2)(x2+2x+4)

Now by using partial fraction method we get,

6x(x2)(x2+2x+4)=Ax2+Bx+Cx2+2x+46x=A(x2+2x+4)+(Bx+C)(x2)......(1)

At x=2,

6(2)=A(22+2(2)+4)+(B(2)+C)(22)A=1

By substituting A=1 in equation (1) we get,

6x=x2+2x+4+Bx22Bx+Cx2C6x=(1+B)x2+(22B+C)x+42C

By Equating LHS and RHS we get,

1+B=0, 22B+C=6 and 42C=0

By solving further we get,

B=1 And C=2.

So by substituting the value of A, B and C we get,

6x(x2)(x2+2x+4)=1x2+(1)x+2x2+2x+4.......

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