Step 1: Start with the integral of \(\ln(x)\) with respect to \(x\): \[ \int \ln(x) \, dx \] Step 2: Use integration by parts, which states: \[ \int u \, dv = uv - \int v \, du \] In this case, we'll choose: \(u = \ln(x)\) and \(dv = dx\)
Step 3: Calculate the differential elements: \[ du = \frac{1}{x} \, dx \quad \text{and} \quad v = \int dx = x \] Step 4: Apply the integration by parts formula: \[ \int \ln(x) \, dx = x\ln(x) - \int x \cdot \frac{1}{x} \, dx \] Step 5: Simplify the integral: \[ \int \ln(x) \, dx = x\ln(x) - \int dx \] Step 6: Evaluate the remaining integral: \[ \int \ln(x) \, dx = x\ln(x) - \int dx = x\ln(x) - x + C \] Where \(C\) is the constant of integration. So, the integral of \(\ln(x)\) with respect to \(x\) is: \[ \int \ln(x) \, dx = x\ln(x) - x + C \]