# Partial fraction question

• Nov 20th 2009, 06:43 AM
Partial fraction question
I need help solving:

integral of (20x-30) / (x(x-2)(x^2-2x+5))

If I try to decompose it;

I will have

LHS from above = (A/x)+(B/x-2) + ?

Stuck here.
• Nov 20th 2009, 06:50 AM
lvleph
You can use the third factor:
$
\frac{20x-30}{x(x-2)(x^2-2x+5)} = \frac{A}{x} + \frac{B}{x-2} + \frac{C}{x^2 -2x+5}
$

C will a function of x.
• Nov 20th 2009, 07:23 AM
TWiX
$
\frac{20x-30}{x(x-2)(x^2-2x+5)} = \frac{A}{x} + \frac{B}{x-2} + \frac{Cx+D}{x^2 -2x+5}
$
• Nov 20th 2009, 07:45 AM
ohh I see, but I didnt think the polynomial could be there under Cx+D because its not factored?

So, now I am suppose to multiply both sides by (x(x-2)(x^2-2x+5))?
• Nov 20th 2009, 12:04 PM
TWiX
Quote:

ohh I see, but I didnt think the polynomial could be there under Cx+D because its not factored?

So, now I am suppose to multiply both sides by (x(x-2)(x^2-2x+5))?

Yes
and find the values of A,B,C and D
substitute and evaluate the resulting integrals.
• Nov 20th 2009, 08:01 PM
simplependulum
$\frac{20x-30}{x(x-2)(x^2-2x+5)}$

$= \frac{20x-30}{(x^2-2x)((x^2-2x)+5)}$

If we let $u = x^2 - 2x$

and we know $\frac{1}{u(u+5)}= \frac{1}{5} \left ( \frac{1}{u} - \frac{1}{ u + 5 } \right )$

The original fraction can be expressed in the form of :

$\frac{20x-30}{5} \left (\frac{1}{u} - \frac{1}{ u + 5 } \right )$

$= (4x - 6) \left ( \frac{ 1}{ x (x-2)} - \frac{1 }{ x^2 - 2x + 5 } \right )$

we can see the last term $\frac{4x-6}{ x^2 -2x +5 }$

the numerator is a polynomial with degree $1$ , that means

we do not need to break it down into pieces anymore .

If we continue , we can obtain

$= \frac{2(2x-3)}{x(x-2)} - \frac{2(2x-3)}{ x^2 -2x +5 }$

$= \frac{ 2[(3/2)(x-2) + (1/2)x] }{x (x-2)} - \frac{2(2x-3)}{ x^2 -2x +5 }$

$= \frac{3}{x} + \frac{1}{x-2} - \frac{2(2x-3)}{x^2 -2x + 5}$
• Nov 21st 2009, 02:53 AM