Explain how to use convolution along with the formula \(\mathcal{L}\{e^{at})=\frac{1}{s-a}\)to find the following inverse Laplace transform:
\[ \mathcal{L}^{-1}\left\{ \frac{7}{2 \, {\left(s + 8\right)} {\left(s + 1\right)}} \right\} \]
Answer:
\[ \frac{1}{2} \, e^{\left(-t\right)} - \frac{1}{2} \, e^{\left(-8 \, t\right)} \]