To solve the integral of the expression [tex]\(\sqrt{x} + \frac{1}{2 \sqrt{x}}\)[/tex] with respect to [tex]\(x\)[/tex], we should follow these steps:
1. Break the integral into two separate integrals:
[tex]\[
\int \left( \sqrt{x} + \frac{1}{2 \sqrt{x}} \right) \, dx = \int \sqrt{x} \, dx + \int \frac{1}{2 \sqrt{x}} \, dx
\][/tex]
2. Solve the first integral [tex]\(\int \sqrt{x} \, dx\)[/tex]:
Recall that [tex]\(\sqrt{x}\)[/tex] can be written as [tex]\(x^{1/2}\)[/tex]. So, we have:
[tex]\[
\int x^{1/2} \, dx
\][/tex]
To integrate [tex]\(x^{1/2}\)[/tex], use the power rule for integration: [tex]\(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\)[/tex] where [tex]\(n \neq -1\)[/tex].
For [tex]\(n = \frac{1}{2}\)[/tex], the integral becomes:
[tex]\[
\int x^{1/2} \, dx = \frac{x^{(1/2)+1}}{(1/2)+1} = \frac{x^{3/2}}{3/2} = \frac{2}{3} x^{3/2}
\][/tex]
3. Solve the second integral [tex]\(\int \frac{1}{2 \sqrt{x}} \, dx\)[/tex]:
Recall that [tex]\(\frac{1}{2 \sqrt{x}}\)[/tex] can be written as [tex]\(\frac{1}{2} x^{-1/2}\)[/tex]. So, we have:
[tex]\[
\int \frac{1}{2} x^{-1/2} \, dx
\][/tex]
Factor out the constant [tex]\(\frac{1}{2}\)[/tex]:
[tex]\[
\frac{1}{2} \int x^{-1/2} \, dx
\][/tex]
To integrate [tex]\(x^{-1/2}\)[/tex], use the power rule for integration similarly:
[tex]\[
\int x^{-1/2} \, dx = \frac{x^{(-1/2)+1}}{(-1/2)+1} = \frac{x^{1/2}}{1/2} = 2 x^{1/2}
\][/tex]
Therefore,
[tex]\[
\frac{1}{2} \int x^{-1/2} \, dx = \frac{1}{2} \cdot 2 x^{1/2} = x^{1/2}
\][/tex]
4. Combine the results of the two integrals:
[tex]\[
\int \left( \sqrt{x} + \frac{1}{2 \sqrt{x}} \right) \, dx = \frac{2}{3} x^{3/2} + x^{1/2}
\][/tex]
5. Write the final answer:
[tex]\[
\int \left( \sqrt{x} + \frac{1}{2 \sqrt{x}} \right) dx = \frac{2}{3} x^{3/2} + \sqrt{x} + C
\][/tex]
where [tex]\(C\)[/tex] is the constant of integration.
So, the integral of [tex]\(\sqrt{x} + \frac{1}{2 \sqrt{x}}\)[/tex] with respect to [tex]\(x\)[/tex] is:
[tex]\[
2 \cdot \frac{x^{3/2}}{3}+ \sqrt{x}
\][/tex]
