How do I find the inverse Laplace Transform of a complex function?

[ns5032]

My goodness.. I have not come across an inverse Laplace transform like this. My teacher let's us just use a chart to figure them out, but this is definitely not on there. How do I find the inverse Laplace transform of:

{ (1/2)+[(5e^-6s)/(4s^2)] } / (s+5)

I already used partial fractions to split up the denominator, so there is one more inverse laplace that I need to do on top of this one, but I figure if I get this one, then I can get the other one as well. Any help??!

 

[exk]

you will need to use the dirac function for the exponential, the rest is pretty standard.

 

[kev08106]

The e^-6s part means that this part of your signal is delayed in time by 6 seconds. so you can do the inverse transform without it and then when you get your time signal for this part, delay it by 6 seconds. So replace t with t-6 in your answer for the time equation and you should be good.

 

[ns5032]

Do the inverse transform without it? Like... just take that whole term out and treat it like it is zero?

 

[Defennder]

You'll need to use this:

$$L(f(t-a)u(t-a)) = e^{-as}F(s)$$

I don't think dirac delta function comes into play here.

 

[sutupidmath]

Or i also think that applying the laplace transforms of integrals would work here.

 

[ForMyThunder]

An easy way to take the inverse Laplace Transform (if you have some knowledge of Complex Calculus) is to take the sum of all the residues of the function e^(zt) f(z), where you are taking the inverse Laplace transform of f(z) and z is the complex variable.