Answer to Question #290046 in Complex Analysis for Aish

Question

Find inverse Laplace transform of F(s) = 50 (s + 1)(s + 5)

Answer 1

$\displaystyle \mathcal{L}(f(t))=F(s)=\frac{50}{(s+1)(s+5)}=\frac{25}{2(s+1)}-\frac{25}{2(s+5)}\\ \Rightarrow f(t)=\mathcal{L}^{-1}F(s)=\mathcal{L}^{-1} \left( \frac{25}{2(s+1)}-\frac{25}{2(s+5)}\right)=\frac{25}{2}\mathcal{L}^{-1}\left(\frac{1}{s+1}\right)-\frac{25}{2}\mathcal{L}^{-1}\left(\frac{1}{s+5}\right)\\ \Rightarrow f(t)=\frac{25}{2}\times e^{-t}-\frac{25}{2}\times e^{-5t}\\ Hence, \\ \mathcal{L}^{-1} \left( \frac{25}{2(s+1)}-\frac{25}{2(s+5)}\right)=\frac{25e^{-t}}{2}-\frac{25e^{-5t}}{2}=\frac{25\left(e^{-t} -e^{-5t}\right)}{2}$