During the winter break, I was reading this book called Street-Fighting Mathematics, which is a cool book teaching tricks and approximations for various calculations. I strongly suggest reading through it if you want to strengthen your mathematical arsenal. In one of the chapters, the reader is invited to give an approximate answer to this integral within 5 minutes, correct to $5\%$:

Let’s look at an algorithmic problem. We have a graph, and we want to color all the vertices with 2 colors such that each edge connects 2 vertices of the same color. This problem is called $\text{2-coloring}$. An example of a 2-colorable graph is shown below.