To integrate the expression \( a^{1/2} - x^{1/2} \), we can write out the integral as follows:
\[ \int (a^{1/2} - x^{1/2}) , dx \]
We can separate the integral into two parts:
\[ \int a^{1/2} , dx - \int x^{1/2} , dx \]
Now, let's integrate each part:
- The first part, \( \int a^{1/2} , dx \), where \( a^{1/2} \) is a constant with respect to \( x \):
\[ \int a^{1/2} , dx = a^{1/2} x \]
- The second part, \( \int x^{1/2} , dx \):
Using the power rule for integration, we have:
\[ \int x^{1/2} , dx = \frac{x^{1/2 + 1}}{1/2 + 1} = \frac{x^{3/2}}{3/2} = \frac{2}{3} x^{3/2} \]
Now, we can combine the results of the two integrals:
\[ \int (a^{1/2} - x^{1/2}) , dx = a^{1/2} x - \frac{2}{3} x^{3/2} + C \]
where \( C \) is the constant of integration.
Thus, the final answer is:
\[ \int (a^{1/2} - x^{1/2}) , dx = a^{1/2} x - \frac{2}{3} x^{3/2} + C \]