Inverse Laplace transformation of a rational function

Question

I just don't seem to be able to crack this inverse Laplace:

$$ \mathcal{L}^{-1} \left\{ \frac{20000s}{s^2 + 20000s + 5\cdot 10^8} \right\} (t). $$

Could someone help me out? I'm totally lost. I don't seem to be able to partial fraction decompose it, so I don't know where to start.

Answer 1

Noting $$ s^2+20000s+5\cdot 10^8=(s+10000)^2+20000^2 $$ one has $$ L^{-1}\bigg(\frac{20000s}{s^2+20000s+5\cdot 10^8}\bigg)=L^{-1}\bigg(\frac{20000(s+10000)-2\cdot10000^2}{(s+10000)^2+20000^2}\bigg)=20000e^{-10000t}\cos(20000t)-10000e^{-10000t}\sin(20000t). $$ Here $$ L^{-1}\bigg(\frac{s-a}{(s-a)^2+b^2}\bigg)=e^{at}\cos(bt), L^{-1}\bigg(\frac{b}{(s-a)^2+b^2}\bigg)=e^{at}\sin(bt).$$

Answer 2

Hint

$(s+10000-20000i)(s+10000+20000i)=s^2+20000s+5\cdot 10^8$