Question1) If $f(x)=\left\{\begin{array}{rr}0 & 0 \leq t<3 \\ t, & t \geq 3\end{array} \quad\right.$, find $\int_{0}^{\infty} f(t) e^{-s t} d t \quad s>0$

Studdy Solution
Combine the results of the integrals from Steps 3 and 10:
$$\int_{0}^{\infty} f(t) e^{-s t} dt = 0 + \left( \frac{3}{s} e^{-3s} + \frac{1}{s^2} e^{-3s} \right)$$
$$\int_{0}^{\infty} f(t) e^{-s t} dt = \left( \frac{3}{s} + \frac{1}{s^2} \right) e^{-3s}$$
Solution: $$\int_{0}^{\infty} f(t) e^{-s t} dt = \left( \frac{3}{s} + \frac{1}{s^2} \right) e^{-3s}$$