\( \int \frac { ( \sqrt { a } - \sqrt { x } ) ^ { 2 } } { \sqrt { x } } \)

To evaluate the integral \[ I = \int \frac{(\sqrt{a} - \sqrt{x})^2}{\sqrt{x}} \, dx, \] we start by expanding the integrand: \[ (\sqrt{a} - \sqrt{x})^2 = a - 2\sqrt{a}\sqrt{x} + x. \] Thus, we can rewrite the integral as: \[ I = \int \frac{a - 2\sqrt{a}\sqrt{x} + x}{\sqrt{x}} \, dx. \] Now, we can separate the integral into three parts: \[ I = \int \frac{a}{\sqrt{x}} \, dx - 2\sqrt{a} \int \frac{\sqrt{x}}{\sqrt{x}} \, dx + \int \frac{x}{\sqrt{x}} \, dx. \] This simplifies to: \[ I = a \int x^{-1/2} \, dx - 2\sqrt{a} \int 1 \, dx + \int x^{1/2} \, dx. \] Now we can compute each integral separately: 1. For \( \int x^{-1/2} \, dx \): \[ \int x^{-1/2} \, dx = 2\sqrt{x} + C_1. \] 2. For \( \int 1 \, dx \): \[ \int 1 \, dx = x + C_2. \] 3. For \( \int x^{1/2} \, dx \): \[ \int x^{1/2} \, dx = \frac{2}{3} x^{3/2} + C_3. \] Now substituting these results back into our expression for \( I \): \[ I = a(2\sqrt{x}) - 2\sqrt{a}(x) + \frac{2}{3} x^{3/2} + C, \] where \( C \) is a constant of integration that combines \( C_1, C_2, \) and \( C_3 \). Thus, we have: \[ I = 2a\sqrt{x} - 2\sqrt{a}x + \frac{2}{3} x^{3/2} + C. \] This is the final result for the integral: \[ \int \frac{(\sqrt{a} - \sqrt{x})^2}{\sqrt{x}} \, dx = 2a\sqrt{x} - 2\sqrt{a}x + \frac{2}{3} x^{3/2} + C. \]