To solve the integral [tex]\(\int (4 \sqrt{x} + 5) \, dx\)[/tex], we need to integrate each term independently.
First, let's rewrite the integral with the integrand separated:
[tex]\[
\int (4 \sqrt{x} + 5) \, dx = \int 4 \sqrt{x} \, dx + \int 5 \, dx
\][/tex]
### Step 1: Integrate [tex]\(4 \sqrt{x}\)[/tex]
Recall that [tex]\(\sqrt{x}\)[/tex] can be written as [tex]\(x^{1/2}\)[/tex]. Therefore, we rewrite the term:
[tex]\[
4 \sqrt{x} = 4 x^{1/2}
\][/tex]
To integrate [tex]\(4 x^{1/2}\)[/tex], we use the power rule for integration:
[tex]\[
\int x^n \, dx = \frac{x^{n+1}}{n+1} + C
\][/tex]
In this case, [tex]\(n = 1/2\)[/tex], so:
[tex]\[
\int 4 x^{1/2} \, dx = 4 \int x^{1/2} \, dx = 4 \left( \frac{x^{1/2 + 1}}{1/2 + 1} \right) + C = 4 \left( \frac{x^{3/2}}{3/2} \right) + C
\][/tex]
Simplify the fraction:
[tex]\[
4 \left( \frac{2}{3} x^{3/2} \right) = \frac{8}{3} x^{3/2}
\][/tex]
### Step 2: Integrate [tex]\(5\)[/tex]
The integral of a constant is straightforward:
[tex]\[
\int 5 \, dx = 5x
\][/tex]
### Step 3: Combine the results
Now we add the results of each integration:
[tex]\[
\int (4 \sqrt{x} + 5) \, dx = \frac{8}{3} x^{3/2} + 5x + C
\][/tex]
Upon reviewing the given answer choices, the closest matching choice based on the integral result is:
[tex]\[
(4) \frac{8}{3} x^{3/2} + 5x
\][/tex]
Therefore, the answer is:
[tex]\[
\boxed{\frac{8}{3} x^{3/2} + 5 x}
\][/tex]
