Let's see some cool application of contour integral first.

1. $\int_\R 1/(1+x^2) dx$

2. $\int_0^\infty 1/(1+x^3) dx$

3. $\int_0^\infty 1/(1+x^3) dx$

4. $\int_\R \frac{x}{x^4+1} e^{-x^2} dx$