Partial Fractional Decomposition to obtain an integral.

[Perseverance]

Homework Statement

Find the exact values of A, B, and C in the following partial fraction decomposition.Then obtain the integral using those values.
1/(x^3-3x^2) = A/x + B/x^2 + C/(x-3)

Homework Equations

I'm not sure at this point.

The Attempt at a Solution

To make the denominator on the right equal to the left...
A(x^2)(x-3) + B(x)(x-3) + C(x)(x^2)
so...
1 = Ax^3 +Cx^3 -3Ax^2 +Bx^2 -3Bx
From cubic values..
A + C = 0 So A= -C
From squared values...
-3A + B = 0 So A = 1/3B which also means C = -1/3B
From x values..
-3B = 0.

Here is where I am at a loss. The x value equation cannot possibly be correct. It would denote that B=0, therefore A and C also equal 0. Am I doing something incorrectly in my multiplication? If not, how is it possible to use the partial fractional decomposition method to make an equation for integration?

 

[zcd]

you made the mistake here: $$1=A(x)(x-3)+B(x-3)+C(x^{2})$$

 

[Perseverance]

Wow, this problem just became possible. Thank you so much :)

 

[Perseverance]

Actually, what implications does factoring out an x have on the numerical answers?

Edit: Never mind I understand how that was able to be done, it should have no implications. thanks again!

 

[HallsofIvy]

Homework Statement

Find the exact values of A, B, and C in the following partial fraction decomposition.Then obtain the integral using those values.
1/(x^3-3x^2) = A/x + B/x^2 + C/(x-3)

Homework Equations

I'm not sure at this point.

The Attempt at a Solution

To make the denominator on the right equal to the left...
A(x^2)(x-3) + B(x)(x-3) + C(x)(x^2)
so...
1 = Ax^3 +Cx^3 -3Ax^2 +Bx^2 -3Bx

Multiplying both sides of your first equation by x^2(x-3) gives
1= Ax(x-3)+ B(x-3)+ Cx^2
Taking x= 0 gives immediately 1= -3B and taking x= 3 gives immediately 1= 9C. Taking x= 1 (just because it is a simple number) gives 1= -2A- 2B+ C. Since you already know B and C, that is easy to solve for A.

From cubic values..
A + C = 0 So A= -C
From squared values...
-3A + B = 0 So A = 1/3B which also means C = -1/3B
From x values..
-3B = 0.

Here is where I am at a loss. The x value equation cannot possibly be correct. It would denote that B=0, therefore A and C also equal 0. Am I doing something incorrectly in my multiplication? If not, how is it possible to use the partial fractional decomposition method to make an equation for integration?