# Help for Partial Fraction

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• Jun 19th 2010, 06:21 AM
moonnightingale
Help for Partial Fraction
Kindly simplify this equation with partial fraction and tell me procedure as well

3/[s(s+1)(s+3)]

Thanks
• Jun 19th 2010, 06:28 AM
simplependulum
$\displaystyle \frac{3}{s(s+1)(s+3) }$

$\displaystyle = \frac{3}{ (s+1) ( s^2 + 3s ) }$

$\displaystyle = \frac{3}{ (s+1) ( (s+1)(s+2) - 2 )}$

$\displaystyle = \frac{1}{2} \left( 3~\frac{s+2 }{ (s+1)(s+2) - 2 } - \frac{3}{ s+1} \right)$

$\displaystyle = \frac{1}{2} \left( 3 ~ \frac{s+2}{ s(s+3) } - \frac{3}{ s+1} \right)$

$\displaystyle = \frac{1}{2} \left( ( \frac{2}{s} + \frac{1}{s+3} ) - \frac{3}{ s+1} \right)$

$\displaystyle = \frac{1}{s} + \frac{1}{2 } \frac{1}{s+3} - \frac{3}{2} ~ \frac{1}{s+1}$
• Jun 19th 2010, 06:43 AM
moonnightingale
Thanks a lot for prompt reply.
How did u perform third step
1/2( 3 ----------)
thanks
• Jun 19th 2010, 06:44 AM
Ackbeet
Or, since all the poles are simple, you could use the Heaviside cover-up method.
• Jun 19th 2010, 06:53 AM
simplependulum
I regard $\displaystyle s+2$ in $\displaystyle (s+1)(s+2) - 2$ as a constant $\displaystyle a$ and let $\displaystyle X = s+1$ so we have

$\displaystyle \frac{2}{X(aX-2 )} = \frac{a}{aX-2} - \frac{1}{X}$

$\displaystyle \frac{1}{X(aX-2)} = \frac{1}{2}( \frac{a}{aX-2} - \frac{1}{X} )$