Let's see some cool application of contour integral first.
1. $\int_\R 1/(1+x^2) dx$
didn't finish, the rest will wait till next week.
2. $\int_0^\infty 1/(1+x^3) dx$
3. $\int_0^\infty e^{-x}/(1+x^3) dx$
4. $\int_\R \frac{x}{x^4+1} e^{-x^2} dx$