# Integrating [1/(x^4 - x)]dx

[1/(x^4 - x)]dx

## The Attempt at a Solution

I factored the denominator to x(x-1)(x^2 + x +1) and I'm not sure if I can use partial fractions.

Gib Z
Homework Helper
Welcome to PF n4rush0.

What exactly is preventing you from carrying out partial fractions? That is the correct way to proceed.

So I set

1/[y(y-1)(y^2+y+1)] = A/y + B/(y-1) + (Cy+D)/(y^2+y+1)

1 = (y-1)(y^2+y+1)A + y(y^2+y+1)B + y(y-1)(Cy+D)

y = 0, A = -1
y = 1, B = 1/3

Not sure what to do for C and D

dextercioby
Homework Helper
You're doing something wrong. Just write the whole polynomial in y and equal it to 1. You should get a system of 4 equations with 4 unknowns which is very easy to solve.

EDIT: You're doing it right, but it's a much more tedious work towards the end than in the method I suggested.

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Gib Z
Homework Helper
You're doing something wrong. Just write the whole polynomial in y and equal it to 1. You should get a system of 4 equations with 4 unknowns which is very easy to solve.

No he's doing it right, it's just a different method. However...

n4rush0: Do you notice the pattern of your substitutions? It's to get some numbers in front of just 1 term, and zero's in front of the rest, right? To find C and D, you need to substitute in the zeros of y^2+y+1. You'll end up with a pair of simultaneous equations for C and D, which you can solve.

However, in this case, since you already know A and B, it's easier at that point of the problem to just sub in simple values for y, say -1 and 2, and solve those.

Alternatively, you can do what bigubau suggested.

Mark44
Mentor
So I set

1/[y(y-1)(y^2+y+1)] = A/y + B/(y-1) + (Cy+D)/(y^2+y+1)

1 = (y-1)(y^2+y+1)A + y(y^2+y+1)B + y(y-1)(Cy+D)

y = 0, A = -1
y = 1, B = 1/3

Not sure what to do for C and D
Out of curiosity, why did you switch from x that was used in your integral to y here?

I tried multiplying it out and got
1 = y^3 (A+B+C) + y^2 (B-C+D) + y(B-D) - A

Even knowing that A = -1 and B = 1/3, I'm still not seeing the "nice" solution.

Oh and, my original problem involved using y, but I wanted to use Mathmatica which solves dx integrals and I copied and pasted [1/(x^4 - x)]dx from there.

dextercioby
Homework Helper
Well, then you've missed the most important point in the partial fraction decomposition method.

on the LHS of what you wrote there's supposed to be the following polynomial in y

0 y^3 + 0 y^2 + 0y +1.