(Solved): 4. The convolution of two peicewise continuous functions \( f \) and \( g \) is defined by \[ (f * ...

4. The convolution of two peicewise continuous functions \( f \) and \( g \) is defined by \[ (f * g)(t)=\int_{0}^{t} f(\tau) g(t-\tau) d \tau \] Suppose that the Laplace transform of \( f \) and \( g \) exist. Then the convolution \( f * g \) properties \[ \mathscr{L}\{f * g\}=\mathscr{L}\{f\} \mathscr{L}\{g\}:=F(s) G(s) \] and \[ \mathscr{L}^{-1}\{F(s) G(s)\}=f * g . \] Let \( m \) and \( n \) be positive integers. Then evaluate the integral \[ \int_{0}^{t}(t-\tau)^{m} \tau^{n} d \tau \]