MHB How can I solve this LaPlace Transform using Laplace Transforms?

[shamieh]

Solve by Laplace Transforms.

So I'm stuck on how to find this $$\mathcal{L}^{-1}$$ $( \frac{\frac{5s}{4} + \frac{13}{4}}{s^2+5s+8} ) $

I'm not sure what t odo. I was thinking I need to use the $\cos(at)$ and $\sin(at)$ formulas but I'm not sure... Any help would be great

 

[shamieh]

nvm, i was doing something wrong

 

[Prove It]

Solve by Laplace Transforms.

So I'm stuck on how to find this $$\mathcal{L}^{-1}$$ $( \frac{\frac{5s}{4} + \frac{13}{4}}{s^2+5s+8} ) $

I'm not sure what t odo. I was thinking I need to use the $\cos(at)$ and $\sin(at)$ formulas but I'm not sure... Any help would be great

$\displaystyle \begin{align*} \frac{\frac{5s}{4} + \frac{13}{4}}{s^2 + 5s + 8} &= \frac{1}{4} \left( \frac{5s + 13}{s^2 + 5s + 8} \right) \\ &= \frac{1}{4} \left[ \frac{5s + 13}{s^2 + 5s + \left( \frac{5}{2} \right) ^2 - \left( \frac{5}{2} \right) ^2 + 8} \right] \\ &= \frac{1}{4} \left[ \frac{5s + 13}{ \left( s + \frac{5}{2} \right) ^2 + \frac{7}{4} } \right] \\ &= \frac{1}{4} \left[ \frac{5 \left( s + \frac{5}{2} \right) + \frac{1}{2}}{\left( s + \frac{5}{2} \right) ^2 + \left( \frac{\sqrt{7}}{2} \right) ^2} \right] \end{align*}$

This is now in a form where you can apply a shift, and then take the transform.