I had a maths exam last Monday and the only question I couldn't do is below. Can anyone shed any light?
Laplace transform the following:
1/(s^2+s+5/4)
Inverse Laplace:
$\displaystyle \mathcal{L} ^{-1} \left\{\frac{1}{s^2+s+\frac{5}{4}}\right\}$
$\displaystyle \implies \mathcal{L}^{-1} \left\{\frac{1}{\left(s^2+s+\frac{1}{4}\right)+\fr ac{5}{4}-\frac{1}{4}}\right\}$
$\displaystyle \implies \mathcal{L}^{-1} \left\{\frac{1}{\left(s+\frac{1}{2}\right)^2+1}\ri ght\}$
Apply the first translation theorem:
$\displaystyle \implies \mathcal{L}^{-1} \left\{\frac{1}{\left(s+\frac{1}{2}\right)^2+1}\ri ght\}$
$\displaystyle \implies f(t)=e^{-\frac{1}{2}t}\sin\left(t\right)$
$\displaystyle \implies \color{red}\boxed{f(t)=\frac{\sin\left(t\right)}{\ sqrt{e^t}}}$