# Thread: Integratin by Partial Fractions

1. ## Integratin by Partial Fractions

Here is what I've done so far...

$\displaystyle (x^3-4x^2-16)/((x^2)(x-4))=A/x+B/x^2+C/(x-4)$

so

$\displaystyle x^3-4x^2-16=A(x^2)(x-4)+B(x)(x-4)+C(x)(x^2)$

=$\displaystyle A(x^3-4x^2)+B(x^2-4x)+C(x^3)$

=$\displaystyle (A+C)(x^3)+(-4A+B)(x^2)+(-4B)(x)$

Now we have to find what the coefficients are equal to, so

A+C=1
-4A+B=-4
B=0

A=1
B=0
C=0

So now I should be able to take the integral of $\displaystyle 1/x+0/x^2+0/(x-4)$

This equals $\displaystyle ln(x)$ on the interval [7,8], so my answer should be $\displaystyle ln(8)-ln(7)$.

This is apparently the wrong answer, so I must have made a mistake somewhere, but I'm not seeing it.

2. Originally Posted by kevin100

Here is what I've done so far...

$\displaystyle (x^3-4x^2-16)/((x^2)(x-4))=A/x+B/x^2+C/(x-4)$

so

$\displaystyle x^3-4x^2-16=A(x^2)(x-4)+B(x)(x-4)+C(x)(x^2)$

=$\displaystyle A(x^3-4x^2)+B(x^2-4x)+C(x^3)$

=$\displaystyle (A+C)(x^3)+(-4A+B)(x^2)+(-4B)(x)$

Now we have to find what the coefficients are equal to, so

A+C=1
-4A+B=-4
B=0

A=1
B=0
C=0

So now I should be able to take the integral of $\displaystyle 1/x+0/x^2+0/(x-4)$

This equals $\displaystyle ln(x)$ on the interval [7,8], so my answer should be $\displaystyle ln(8)-ln(7)$.

This is apparently the wrong answer, so I must have made a mistake somewhere, but I'm not seeing it.
if the degree of the numerator is greater than or equal to the degree of the denominator, we must divide before doing partial fractions.

the highest power in the top is the same as that of the bottom. thus we must first divide the numerator by the denominator by long division of polynomials, or alternatively, note that

$\displaystyle \frac {x^3 - 4x^2 - 16}{x^3 - 4x^2} = \frac {x^3 - 4x^2}{x^3 - 4x^2} - \frac {16}{x^3 - 4x^2} = 1 - \frac {16}{x^3 - 4x^2}$

so your integral is $\displaystyle \int_7^8 \bigg( 1 - \frac {16}{x^3 - 4x^2} \bigg)~dx$

integrating 1 is easy, use partial fractions for the latter part

3. I'm still not getting it. My work doesn't seem to make much sense so I wont bother typing it out.

4. Originally Posted by kevin100
I'm still not getting it. My work doesn't seem to make much sense so I wont bother typing it out.
$\displaystyle \frac {16}{x^2(x - 4)} = \frac Ax + \frac B{x^2} + \frac C{x - 4}$

$\displaystyle \Rightarrow 16 = Ax(x - 4) + B(x - 4) + Cx^2$ ...............(1)

plugging in $\displaystyle x = 0$ in (1) we get $\displaystyle \boxed{B = -4}$

plugging in $\displaystyle x = 4$ in (1) we get $\displaystyle \boxed{C = 1}$

plugging in $\displaystyle x=1,~B=-4,$ and $\displaystyle C = 1$ in (1) we get $\displaystyle \boxed{A = -1}$

5. Here.