Let \(u = \ln x\) and \(dv = dx\). Then \(du = \frac{1}{x} \, dx\) and \(v = x\).

Using integration by parts, \(\int u \, dv = uv - \int v \, du\).

Substitute the values: \(\int \ln x \, dx = x \ln x - \int x \cdot \frac{1}{x} \, dx\).

This simplifies to \(x \ln x - \int 1 \, dx = x \ln x - x + C\).

Evaluate from 2 to 4:

\(\left[ x \ln x - x \right]_{2}^{4} = (4 \ln 4 - 4) - (2 \ln 2 - 2)\).

Calculate each term:

\(4 \ln 4 - 4 = 4 \cdot 2 \ln 2 - 4 = 8 \ln 2 - 4\).

\(2 \ln 2 - 2 = 2 \ln 2 - 2\).

Combine the results:

\((8 \ln 2 - 4) - (2 \ln 2 - 2) = 8 \ln 2 - 4 - 2 \ln 2 + 2\).

\(= 6 \ln 2 - 2\).