Find $ \int {dt\over 2t+1}$ a simple question, but I'm stuck anyway:
How to integrate this:
$$ \int {dt\over 2t+1}  = ? $$
Is it simply: $\ln|2t +1| $
or do I need Chain rule like: $$\ln|2t +1| \cdot \frac{d}{dt}(2t + 1) $$

This should be the straight-cut way to integrate the expression.
As Cameron Buie responded, let $u = 2t + 1$.  Then, $du = 2 dt$.  Making that substitution for the integral, we have:
$$\int \frac{du}{2u}$$
$$= \frac{1}{2} \int \frac{du}{u}$$
$$= \frac{1}{2} \ln|u| + \mathrm{c}$$
$$= \frac{1}{2} \ln|2t + 1| + \mathrm{c}$$
