Solve the integral integral (2 x)/(16 + 2 x) dx.

Let $u = \cot x$.

Differentiate $u$ with respect to $x$:

So, $du = -\csc^2 x \, dx$, which means $\csc^2 x \, dx = -du$.

The integral becomes:

This integral is in the form $\int \frac{1}{a^2 + x^2} \, dx = \frac{1}{a} \arctan\left(\frac{x}{a}\right) + C$.
Here, $a^2 = 16$, so $a = 4$.