# Need help finding some unknown constants

## Homework Statement

I have an integral that I'm trying to split into partial fractions, and I've gotten to an equation but I'm not sure how to solve this one.

## The Attempt at a Solution

93x+2 = Ax^2 + Bx + A - 4B

I have no idea, because I usually don't have a x^2 term to deal with. I assumed that A should be 0 because of this, but that requires B to equal 93, and for A - 4B to equal 2. So, that's not the right direction.

Yes A = 0

and A - 4B = 2

You know what A is...

How does that work? That means B is - 1/2, and

93x + 2 = -1/2x + 2

Is not true.

SammyS
Staff Emeritus
Homework Helper
Gold Member
Yes A = 0

and A - 4B = 2

You know what A is...
And this requires B = -1/2 , but that contradicts B = 93.

So, there appears to be something wrong with the way you obtained your equation: 93x+2 = Ax2 + Bx + A - 4B.

Show us how you came up with that so we can help you.

Okay

$\int \frac{93x+2}{x^{3}-4x^{2}+x-4}dx$

I factored this by finding the factor of 4 that yields a 0 when plugged into the denominator, which is 4 itself. So I concluded one term is (x-4), and found the other through polynomial long division.

$\int \frac{93x+2}{(x-4)(x^{2}+1)}dx$

So I then attempted to split this into partial fractions.

$\int \frac{A}{(x-4)}+ \frac{B}{(x^{2}+1)}dx$

So then

93x + 2 = Ax^2 + A + Bx - 4B

What's wrong with the equation?

SammyS
Staff Emeritus
Homework Helper
Gold Member
That needs to be $\displaystyle \frac{A}{x-4}+ \frac{Bx + C}{x^{2}+1}$

Oh, ok. Why is that? Because I have a variable in my numerator?

SammyS
Staff Emeritus
Homework Helper
Gold Member
The denominator x2 + 1 is quadratic in x, so the most general numerator is linear in x.

When working it I get:

A = 23.25
B = - A
C = 5.3125

Agree?

SammyS
Staff Emeritus
Homework Helper
Gold Member
No, although, B = -A is correct.

Did you get

93x + 2 = A(x2 + 1) +(Bx +C)(x - 4) ?

Hint: Set x = 4 to find A.

I get

x^2(A+B) + x(C-4B) - 4C + A = 93x + 2

From there, I said that A+B = 0, C-4B = 93, and -4C + A = 2 and I am getting it wrong every time.