# Thread: Help for Partial Fraction

1. ## Help for Partial Fraction

Kindly simplify this equation with partial fraction and tell me procedure as well

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

Thanks

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

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

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

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

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

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

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

3. Thanks a lot for prompt reply.
How did u perform third step
1/2( 3 ----------)
thanks

4. Or, since all the poles are simple, you could use the Heaviside cover-up method.

5. 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

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

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