6. In this problem, we shall learn how to compute ∫ 0 + ∞ sint t dt via complex integration. Let us first define I(r) = ∫ |z| = r z −1 e iz dz. (a) (8 pts) Show that I(r) → 0 as r → ∞. (b) (10 pts) Show that I(r) → π i as r → 0. (c) (2 pts) For 0 < s < r < ∞, consider a path γ(s, t) = [s, r] + γr + [−r, − s] − γs where γr(t) = re it and γs(t) = se it on t ∈ [0, π]. What is ∫ γ z −1 e it dz? (d) (10 pts) From (a)-(c), evaluate ∫ 0 + ∞ sint t dt. (Hint: t ↦ sint/t is an even function.)