1. The problem is to find the integral of $\ln x$ with respect to $x$, i.e., $\int \ln x \, dx$. 2. We use integration by parts formula: $$\int u \, dv = uv - \int v \, du$$ 3. Let $u = \ln x$ and $dv = dx$. Then, $du = \frac{1}{x} dx$ and $v = x$. 4. Substitute into the formula: $$\int \ln x \, dx = x \ln x - \int x \cdot \frac{1}{x} dx = x \ln x - \int 1 \, dx$$ 5. Simplify the integral: $$\int 1 \, dx = x$$ 6. Therefore, $$\int \ln x \, dx = x \ln x - x + C$$ where $C$ is the constant of integration. This is the final answer.