1. ## Partial Fractions

I am in the middle of solving a integro-differential, and I need to use inverse Laplace transforms to complete the last bit, I have the following:

$
x(s)= \frac{1}{s\left[1+\frac{s}{s+1}\right]}+ \frac{1}{s^2\left[1+\frac{s}{s+1}\right]}-\frac{1}{s+1\left[1+\frac{s}{s+1}\right]}
$

I have simplified this to the following form:

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

Then using partial fractions I have taken the above term by term, and done the following:

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

Second term:

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

Can some please varify if I have used partial fractions correctly on the first 2 expressions

Also I do not understand how to use partial fraction on the third expression, can some please help and explain how I would do this... ?

2. Originally Posted by ubhik
I am in the middle of solving a integro-differential, and I need to use inverse Laplace transforms to complete the last bit, I have the following:

$
x(s)= \frac{1}{s\left[1+\frac{s}{s+1}\right]}+ \frac{1}{s^2\left[1+\frac{s}{s+1}\right]}-\frac{1}{s+1\left[1+\frac{s}{s+1}\right]}
$

I have simplified this to the following form:

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

Then using partial fractions I have taken the above term by term, and done the following:

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

Second term:

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

Can some please varify if I have used partial fractions correctly on the first 2 expressions

Also I do not understand how to use partial fraction on the third expression, can some please help and explain how I would do this... ?
You have.

(s+1)^2+s to get s^2 + 3s + 1. This factorises, but not nicely.

3. Originally Posted by ubhik
$
x(s)= \frac{s+1}{s(s-2)}+\frac{s+1}{s^2(2s+1)}-\frac{s+1}{(s+1)^2+s}
$

Also I do not understand how to use partial fraction on the third expression, can some please help and explain how I would do this... ?
Break it in a different way:

$\frac{s+1}{s^2+3s+1} = \frac{s+1}{(s+\frac32)^2 - \frac54}$