1. **State the problem:** We need to solve the integral $$\int \frac{1}{\ln x} \, d(\ln x)$$. 2. **Understand the integral:** The integral is with respect to $d(\ln x)$, which means the variable of integration is $\ln x$. 3. **Substitute:** Let $u = \ln x$. Then $d(\ln x) = du$. 4. **Rewrite the integral:** The integral becomes $$\int \frac{1}{u} \, du$$. 5. **Recall the integral formula:** $$\int \frac{1}{u} \, du = \ln |u| + C$$. 6. **Back-substitute:** Replace $u$ with $\ln x$ to get $$\ln |\ln x| + C$$. 7. **Conclusion:** The solution to the integral is $$\boxed{\ln |\ln x| + C}$$. This corresponds to option b).