Sorry that the camera has trouble focusing the first 2 minutes. (0:00) Plan for the last 2 weeks. (2:11) Consider whether God's plan for your life could include exploring complex analysis further. (3:26) Antiderivative of 1/(1+x^2) and graphs. (6:10) Area under the graph of 1/(1+x^2) evaluated as an improper integral which is defined as the sum of two improper integrals (and these integrals do converge). (10:39) Integrate 1/(1+z^2) over various closed contours using Mathematica to visualize the corresponding vector fields.
Based on "Fundamentals of Complex Analysis, with Applications to Engineering and Science", by E.B. Saff and A.D. Snider (3rd Edition). "Visual Complex Analysis", by Tristan Needham, is also referred to a lot. Mathematica is often used, especially to visualize complex analytic (conformal) mappings.