Inverse Laplace of a function

Question

I am really searching for hours now for the inverse laplace transformation of the following function:

$$\frac{75s + 12739.726}{s( 0.0365s^2 + 81.2s + 12739.726)}$$

If I put this in WolframAlpha the solution needs to be:

$$1-e^{-2054.79t}-2.11263\times10^{-10}e^{-169.863t}$$

This is a step response of an electrical system.. It is a long time ago I used laplace etc. and some help would be really useful.

Thanks in advance.

Answer 1

$$\frac{75s + 12739.726}{s( 0.0365s^2 + 81.2s + 12739.726)} = \frac 1 s -\frac{ 2.11263 \times 10^{-10}}{s + 169.863} - {1 \over s + 2052.79} $$

Now the inverse Laplace transform of $a \over s + b$ is $ae^{-bt}$ and the inverse Laplace transform of $1/s$ is the identity function (or the Heavyside function).

E.g., $$\mathcal L[ae^{-bt}](s) = a \int_0^\infty e^{-bt}e^{-st} \ dt = \frac{a}{s+b}$$

Does that answer your question?