How to use different method to compute Laplace inverse

Question

I was trying to find the inverse of the Laplace transform, which is $L^{-1}(\frac{3s+5}{s^2(s^2+4)})$. I used Partial Fractions to compute and I got $\frac{3}{4}+\frac{5}{4}t-\frac{3}{4}\cos(2t)-\frac{5}{8}\sin(2t)$. I wonder if there are other methods that can solve it. I tried to use convolution but couldn't get the same result. Really appreciate your help!

Answer 1

We have

$$ \frac{3s+5}{s^2(s^2+4)} = \frac 12\left(\frac{2}{s^2+4}\right)\left(\frac 3s+\frac {5}{s^2}\right) $$

then

$$ \mathcal{L}^{-1}\left[\frac{3s+5}{s^2(s^2+4)}\right]=\frac 12\int_0^t\sin(2\tau)\left(3+5(t-\tau)\right)d\tau = \frac{3}{4}+\frac{5 t}{4}-\frac{5}{8} \sin (2 t)-\frac{3}{4} \cos (2 t) $$