Question

intergrate a^1/2 - x^1/2

Answer 1

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:

1. The first part, \( \int a^{1/2} \, dx \), where \( a^{1/2} \) is a constant with respect to \( x \):

\[
\int a^{1/2} \, dx = a^{1/2} x
\]

2. 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:

\[
\int (a^{1/2} - x^{1/2}) \, dx = a^{1/2} x - \frac{2}{3} x^{3/2} + C
\]