1. Aplicamos propiedades de logaritmos: $\log(\sqrt{x}) = \log(x^{1/2}) = \frac{1}{2} \log x$.
$$\frac{1}{2} \int_{1}^e \log x \, dx$$
2. Integramos por partes ($\int u \, dv = uv - \int v \, du$): sea $u = \log x \implies du = \frac{1}{x}dx$; $dv = dx \implies v = x$.
$$\frac{1}{2} [x \log x - x]_1^e$$
3. Evaluamos:
$$\frac{1}{2} [(e \cdot 1 - e) - (1 \cdot 0 - 1)] = \frac{1}{2} [0 - (-1)] = \frac{1}{2}$$