# Help for Partial Fraction

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• June 19th 2010, 07:21 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
• June 19th 2010, 07:28 AM
simplependulum
$\frac{3}{s(s+1)(s+3) }$

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

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

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

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

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

$= \frac{1}{s} + \frac{1}{2 } \frac{1}{s+3} - \frac{3}{2} ~ \frac{1}{s+1}$
• June 19th 2010, 07:43 AM
moonnightingale
Thanks a lot for prompt reply.
How did u perform third step
1/2( 3 ----------)
thanks
• June 19th 2010, 07:44 AM
Ackbeet
Or, since all the poles are simple, you could use the Heaviside cover-up method.
• June 19th 2010, 07:53 AM
simplependulum
I regard $s+2$ in $(s+1)(s+2) - 2$ as a constant $a$ and let $X = s+1$ so we have

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

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