Integration by parts for \int \ln x dx :

Solution:
1. Identify the components of integration by parts:
- Let u = \ln x, then du = \frac{1}{x} \, dx.
- Let dv = dx, then v = x.

2. Apply the integration by parts formula \int u \, dv = uv - \int v \, du:
* The formula becomes:
\int \ln x \, dx = x \ln x - \int x \left(\frac{1}{x} \right) \, dx.

3. Simplify the remaining integral:
* \int x \left(\frac{1}{x} \right) \, dx = \int 1 \, dx = x + C, where C is the integration constant.

4. Substitute back to get the final result:
* \int \ln x \, dx = x \ln x - x + C.