Help me solve $\int \ln(2x+1)dx$

Hint. Just write $$ \frac {2x} {2x+1}=\frac {2x+1-1} {2x+1}=1-\frac {1} {2x+1} $$ and integrate each term.

you can integrate $$\int \frac{2x}{2x+1} \, dx = \int \left(1 - \frac1{2x+1}\right) \, dx = x - \frac 12 \ln|2x+1| + C$$

p.s. i think it would have been easier had you made the substitution $u = 2x+1$ at the very beginning. this trick works on integrals involving composition with $ax + b.$ here are some examples $\int \dfrac{1}{(ax+b)^2} \, dx, \int \sin (ax + b) \, dx, \int \ln (ax + b)\, dx, etc.$