First, rewrite the integrand: \(\sqrt{x} = x^{1/2}\) and \(\frac{2}{\sqrt{x}} = 2x^{-1/2}\).
Integrate each term separately:
\(\int x^{1/2} \, dx = \frac{x^{3/2}}{3/2} = \frac{2}{3}x^{3/2}\).
\(\int 2x^{-1/2} \, dx = 2 \cdot \frac{x^{1/2}}{1/2} = 4x^{1/2}\).
Combine the integrals: \(\frac{2}{3}x^{3/2} + 4x^{1/2}\).
Evaluate from 1 to 4:
\(\left[ \frac{2}{3}x^{3/2} + 4x^{1/2} \right]_{1}^{4} = \left( \frac{2}{3}(4)^{3/2} + 4(4)^{1/2} \right) - \left( \frac{2}{3}(1)^{3/2} + 4(1)^{1/2} \right)\).
Calculate each part:
\((4)^{3/2} = 8\) and \((4)^{1/2} = 2\).
\(\frac{2}{3} \times 8 + 4 \times 2 = \frac{16}{3} + 8 = \frac{40}{3}\).
\(\frac{2}{3} \times 1 + 4 \times 1 = \frac{2}{3} + 4 = \frac{14}{3}\).
Subtract: \(\frac{40}{3} - \frac{14}{3} = \frac{26}{3}\).
