Answer
The solution is $$\int\frac{\ln x}{x(3+2\ln x)}dx=\frac{1}{4}(3+2\ln x)-\frac{3}{4}\ln|3+2\ln x|+c.$$
Work Step by Step
To solve the integral
$$\int\frac{\ln x}{x(3+2\ln x)}dx$$
we will use substitution $\ln x=t$ which gives us $\frac{dx}{x}=dt$. Putting this into the integral we have:
$$\int\frac{\ln x}{x(3+2\ln x)}dx=\int\frac{t}{3+2t}dt.$$
Now we will use new substitution $z=3+2t\Rightarrow\frac{dz}{2}=dt,t=\frac{z-3}{2}.$ Using this we get:
$$\int\frac{t}{3+2t}dt=\frac{1}{2}\int\frac{\frac{z-3}{2}}{z}dz=\frac{1}{4}\int\frac{z-3}{z}dz=\frac{1}{4}\int dz-\frac{3}{4}\int\frac{dz}{z}=\frac{1}{4}z-\frac{3}{4}\ln|z|+c.$$
We will express $z$ in terms of $t$ and then $t$ in terms of $x$:
$$\frac{1}{4}z-\frac{3}{4}\ln|z|+c=\frac{1}{4}(3+2t)-\frac{3}{4}\ln|3+2t|+c=\frac{1}{4}(3+2\ln x)-\frac{3}{4}\ln|3+2\ln x|+c.$$
