A common integral that needs to be taken for mechanics problems in physics forces in the surface integral. The issue lies in that it is not always feasible to convert cartesian into polar coordinates, and solves this problem easily. What I am hoping to do is figure out a streamlined substitution to solve this integral.
$$2\pi \int x^2 \sqrt{1+4x^2}~\mathrm dx $$
Try $x= \frac{\tan(t)}{2}$. In general, if you have a $1+bx^2$ under a radical, you want to go with $\frac{1}{\sqrt{b}}\tan(t)=x$.