# Help on Partial Fractions

• Jan 5th 2010, 01:22 AM
Paymemoney
Help on Partial Fractions
Hi
The following questions i am having trouble finding the partial fractions to them:

1)$\displaystyle \frac{x^2+x-1}{(x^2+1)^2}$

2)Can someone tell me if i have done this correctly.

$\displaystyle \frac{x^2-1}{x^2-5x+6}$
Do Long Division and i get this:$\displaystyle 1+\frac{-5x+7}{(x-3)(x-2)}$
$\displaystyle \frac{A}{(x-3)}+\frac{B}{(x-2)}$

$\displaystyle A(x-2)+B(x-3)=-5x+7$

$\displaystyle A+B=-5$

$\displaystyle A=-5-B$

$\displaystyle -2A-3B=7$

$\displaystyle -2(-5-B)-3B=7$

$\displaystyle 10-B=7$

$\displaystyle B=3$

$\displaystyle A=-5-3$

$\displaystyle A=-8$

therefore: $\displaystyle 1-\frac{8}{(x-3)}+\frac{3}{(x-2)}$

now answers says its $\displaystyle 1-\frac{3}{(x-2)}+\frac{8}{(x-3)}$.
i have tried doing this equation many times i don't understand how they got the above answer.

P.S
• Jan 5th 2010, 01:33 AM
dedust
Quote:

Originally Posted by Paymemoney
Hi
The following questions i am having trouble finding the partial fractions to them:

1)$\displaystyle \frac{x^2+x-1}{(x^2+1)^2}$

2)Can someone tell me if i have done this correctly.

$\displaystyle \frac{x^2-1}{x^2-5x+6}$
Do Long Division and i get this:$\displaystyle 1+\frac{-5x+7}{(x-3)(x-2)}$ $\displaystyle \leftarrow$ should be $\displaystyle \frac{x^2-1}{x^2-5x+6} = \frac{x^2-5x + 6 + 5x -7}{x^2-5x+6} = 1+\frac{5x-7}{(x-3)(x-2)}$

. . .
• Jan 5th 2010, 02:02 AM
Prove It
$\displaystyle \int{\frac{x^2 + x - 1}{(x^2 + 1)^2}\,dx}$.

Let $\displaystyle x = \tan{\theta}$ so that $\displaystyle dx = \sec^2{\theta}\,d\theta$.

The integral becomes

$\displaystyle \int{\frac{\tan^2{\theta} + \tan{\theta} - 1}{(\tan^2{\theta} + 1)^2}\,\sec^2{\theta}\,d\theta}$

$\displaystyle = \int{\frac{\tan^2{\theta} + \tan{\theta} - 1}{(\sec^2{\theta})^2}\,\sec^2{\theta}\,d\theta}$

$\displaystyle = \int{\cos^2{\theta}(\tan^2{\theta} + \tan{\theta} - 1)\,d\theta}$

$\displaystyle = \int{\sin^2{\theta} + \sin{\theta}\cos{\theta} - 1\,d\theta}$

$\displaystyle = \int{\sin{\theta}\cos{\theta} - \cos^2{\theta}\,d\theta}$

$\displaystyle = \int{\frac{1}{2}\sin{(2\theta)} - \frac{1}{2}\cos{(2\theta)} - \frac{1}{2}\,d\theta}$

$\displaystyle = -\frac{1}{4}\cos{(2\theta)} - \frac{1}{4}\sin{(2\theta)} - \frac{1}{2}\theta + C$

$\displaystyle = -\frac{1}{4}\left(\frac{1 - \tan^2{\theta}}{1 + \tan^2{\theta}}\right) - \frac{1}{4}\left(\frac{2\tan{\theta}}{1 + \tan^2{\theta}}\right) - \frac{1}{2}\theta + C$

$\displaystyle = -\frac{1}{4}\left(\frac{1 - x^2}{1 + x^2}\right) - \frac{1}{4}\left(\frac{2x}{1 + x^2}\right) - \frac{1}{2}\arctan{x} + C$

$\displaystyle = \frac{x^2 - 2x - 1}{4(1 + x^2)} - \frac{1}{2}\arctan{x} + C$.
• Jan 5th 2010, 02:07 AM
Paymemoney
for the first question i want to know how to find the partial fraction not intergrating it.
• Jan 5th 2010, 02:12 AM
Paymemoney
Quote:

Originally Posted by dedust
. . .

ok i understand now, but is there a way of knowing which value you divide by? In this question i divided top into bottom and it was wrong, however when you divide bottom into top you get 5x-7.
• Jan 5th 2010, 03:00 AM
HallsofIvy
Quote:

Originally Posted by Paymemoney
ok i understand now, but is there a way of knowing which value you divide by? In this question i divided top into bottom and it was wrong, however when you divide bottom into top you get 5x-7.

When did you ever learn to divide the denominator of a fraction by the numerator? $\displaystyle \frac{a}{b}$ means a divided by b! You are correct that Paymethemoney's division is wrong. It should be $\displaystyle 1+ \frac{5x- 7}{x^2- 5x+ 1}$. He has the signs in the remainder wrong.

The point in the second problem is that you only use partial fractions on "proper fractions". That's why you need to divide first.

For the first problem, $\displaystyle \frac{x^2+ x- 1}{(x^2+ 1)^2}$, because the denominator is an irreducible quadratic, squared, we need
$\displaystyle \frac{x^2+ x- 1}{(x^2+ 1)^2}= \frac{Ax+ B}{x^2+ 1}+ \frac{Cx+ D}{(x^2+1)^2}$.
• Jan 5th 2010, 03:06 AM
dedust
Quote:

Originally Posted by HallsofIvy
It should be $\displaystyle 1+ \frac{5x- 7}{x^2- 5x+ 1}$.

a small correction in the denominator

$\displaystyle 1+ \frac{5x- 7}{x^2- 5x+ 6}$
• Jan 5th 2010, 02:56 PM
Paymemoney
well this is how i done my long division by divide $\displaystyle x^2-1$into $\displaystyle x^2-5x+6$

someone tell me where i have gone wrong:
http://img130.imageshack.us/img130/4717/divisionf.jpg

P.S
• Jan 5th 2010, 03:24 PM
ANDS!
The only difference is a distributed -1.
• Jan 5th 2010, 03:33 PM
dedust
Quote:

Originally Posted by Paymemoney
well this is how i done my long division by divide $\displaystyle x^2-1$into $\displaystyle x^2-5x+6$

someone tell me where i have gone wrong:
http://img130.imageshack.us/img130/4717/divisionf.jpg

P.S

nothing wrong,..

but this is $\displaystyle \frac{x^2 - 5x + 6}{x^2 - 1}$
• Jan 6th 2010, 01:40 AM
Paymemoney
Quote:

Originally Posted by HallsofIvy
For the first problem, $\displaystyle \frac{x^2+ x- 1}{(x^2+ 1)^2}$, because the denominator is an irreducible quadratic, squared, we need
$\displaystyle \frac{x^2+ x- 1}{(x^2+ 1)^2}= \frac{Ax+ B}{x^2+ 1}+ \frac{Cx+ D}{(x^2+1)^2}$.

i tired doing this method however when i found the values it did not match the book's answers.
Here's what i have done:

$\displaystyle \frac{x^2+x-1}{(x^2+1)^2}$

$\displaystyle \frac{Ax+B}{(x^2+1)}+\frac{Cx+D}{(x^2+1)^2}$

$\displaystyle (Ax+B)(x^2+1)(x^2+1)+(Cx+D)(x^2+1)=x^2+x-1$

The coefficients
$\displaystyle A+B=0$
$\displaystyle 2A+C=0$
$\displaystyle 2B+D=1$
$\displaystyle A+C=1$
$\displaystyle B+D=-1$

Find B using simultaneous equations:
(1)$\displaystyle 2B+D=1$
(2)$\displaystyle B+D=-1$
(1)-(2)=
$\displaystyle B=2$

Find D
$\displaystyle B+D=-1$
$\displaystyle 2+D=-1$
$\displaystyle D=-3$

Find A
$\displaystyle A+B=0$
$\displaystyle A+2=0$
$\displaystyle A=-2$

Find C
$\displaystyle A+C=1$
$\displaystyle -2+C=1$
$\displaystyle C=3$
The book's answer is$\displaystyle \frac{1}{x^2+1}+\frac{x-2}{(x^2+1)^2}$
• Jan 6th 2010, 02:51 AM
dedust
Quote:

Originally Posted by Paymemoney
i tired doing this method however when i found the values it did not match the book's answers.
Here's what i have done:

$\displaystyle \frac{x^2+x-1}{(x^2+1)^2}$

$\displaystyle \frac{Ax+B}{(x^2+1)}+\frac{Cx+D}{(x^2+1)^2}$

$\displaystyle (Ax+B)(x^2+1)(x^2+1)+(Cx+D)(x^2+1)=x^2+x-1$ $\displaystyle \rightarrow$ should be $\displaystyle (Ax+B)(x^2+1)+(Cx+D)=x^2+x-1$

. . .
• Jan 6th 2010, 03:00 AM
Prove It
Quote:

Originally Posted by Paymemoney
for the first question i want to know how to find the partial fraction not intergrating it.

Then you should have put this in Algebra or Pre-Calculus.

The only reason I can think of to use Partial Fractions in Calculus would be to find the integral.