Exercise 12

Let $f(t)$ be a peicewise continuous function for $t \geq 0$. In addition, suppose that $f(t)$satisfies the condition $|f(t)| \leq M e^{ct}$ with $t \geq 0$.

1. Prove that $${\mathcal L} \left( \int_0^t f(\tau) \, d\tau \right) = \frac{1}{s} {\mathcal L}(f(t) = \frac{F(s)}{s}.$$

2. Show that $${\mathcal L}^{-1} \left( \frac{F(s)}{s} \right) = \int_0^t f(\tau) \, d\tau$$ follows from (a).