Understanding how to compute $\int_{0}^{+\infty} \frac{\sin t}{t}dt $ via complex integration.

Question

Question: Understanding how to compute $\int_{0}^{+\infty} \frac{\sin t}{t}dt $ via complex integration.

Let us first define $ I(r)=\int_{(|z|=r)} \frac{e^{iz}}{z}dz. $

(a) Show that $I(r) \to 0 $ as $ r \to \infty. $

(b) Show that $I(r) \to \pi i$ as $ r \to 0. $

(c) Consider a path $ \gamma (s,t) = [s,r]+ \gamma_{r} + [-r,-s] - \gamma_{s} $ for $ 0<s<r<\infty $ where $ \gamma_{r}(t) = re^{it}$ and $\gamma_{s}(t)=se^{it}$ on $t$ in $ [0,\pi ].$ What is $\int_{(|z|=r)} z^{-1}e^{iz}dz ? $

(d) From (a)-(c), evaluate $\int_0^{+\infty} \frac{\sin t}{t}dt.$ (Hint: $t \mapsto \frac{\sin t}{t}$ is an even function.)

attempt: for a) If $r$ becomes very large, $1/z$ becomes very small. Also, $ \max|e^{iz}|$ is $1$ so $I(r) \le 1/r$ and for b) we can first simplify the integral by using talor series but then I dont know what to do next. Is there any other conditions that I need to consider? For c), do we evaluate the integral by seperating it first? (FYI I have not learned about Residue Theorem) Any help is appreciated.