1. The problem is to find the integral of $\ln(2x)$ with respect to $x$. 2. We use the integration formula for logarithmic functions: $$\int \ln(ax)\,dx = x\ln(ax) - x + C$$ where $a$ is a constant and $C$ is the constant of integration. 3. Applying this formula, let $a=2$, so: $$\int \ln(2x)\,dx = x\ln(2x) - x + C$$ 4. To verify, differentiate the result: $$\frac{d}{dx} \left(x\ln(2x) - x\right) = \ln(2x) + x \cdot \frac{1}{2x} \cdot 2 - 1 = \ln(2x) + 1 - 1 = \ln(2x)$$ 5. Thus, the integral is: $$\boxed{\int \ln(2x)\,dx = x\ln(2x) - x + C}$$