To evaluate the indefinite integral [tex]\(\int \left( \frac{5}{\sqrt{x}} + 5 \sqrt{x} \right) \, dx\)[/tex], let's break down the integral into two parts and solve each part separately.
The integral can be written as:
[tex]\[
\int \left( \frac{5}{\sqrt{x}} + 5 \sqrt{x} \right) \, dx = \int \frac{5}{\sqrt{x}} \, dx + \int 5 \sqrt{x} \, dx
\][/tex]
### Evaluate [tex]\(\int \frac{5}{\sqrt{x}} \, dx\)[/tex]
First, let's simplify [tex]\(\frac{5}{\sqrt{x}}\)[/tex]:
[tex]\[
\frac{5}{\sqrt{x}} = 5 x^{-\frac{1}{2}}
\][/tex]
Now we can integrate [tex]\(5 x^{-\frac{1}{2}}\)[/tex]:
[tex]\[
\int 5 x^{-\frac{1}{2}} \, dx
\][/tex]
Use the power rule of integration, which states [tex]\(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\)[/tex] for [tex]\(n \neq -1\)[/tex]:
[tex]\[
n = -\frac{1}{2}
\][/tex]
[tex]\[
\int x^{-\frac{1}{2}} \, dx = \frac{x^{-\frac{1}{2} + 1}}{-\frac{1}{2} + 1} = \frac{x^{\frac{1}{2}}}{\frac{1}{2}}
\][/tex]
Multiply by the constant 5:
[tex]\[
\int 5 x^{-\frac{1}{2}} \, dx = 5 \cdot 2 x^{\frac{1}{2}} = 10 \sqrt{x}
\][/tex]
### Evaluate [tex]\(\int 5 \sqrt{x} \, dx\)[/tex]
First, let's rewrite [tex]\(\sqrt{x}\)[/tex] using exponents:
[tex]\[
\sqrt{x} = x^{\frac{1}{2}}
\][/tex]
Now we can integrate [tex]\(5 x^{\frac{1}{2}}\)[/tex]:
[tex]\[
\int 5 x^{\frac{1}{2}} \, dx
\][/tex]
Use the power rule of integration:
[tex]\[
n = \frac{1}{2}
\][/tex]
[tex]\[
\int x^{\frac{1}{2}} \, dx = \frac{x^{\frac{1}{2} + 1}}{\frac{1}{2} + 1} = \frac{x^{\frac{3}{2}}}{\frac{3}{2}}
\][/tex]
Multiply by the constant 5:
[tex]\[
\int 5 x^{\frac{1}{2}} \, dx = 5 \cdot \frac{2}{3} x^{\frac{3}{2}} = \frac{10}{3} x^{\frac{3}{2}}
\][/tex]
### Combine the results
Adding the results of both integrals:
[tex]\[
\int \left( \frac{5}{\sqrt{x}} + 5 \sqrt{x} \right) \, dx = 10 \sqrt{x} + \frac{10}{3} x^{\frac{3}{2}} + C
\][/tex]
To present the final answer neatly:
[tex]\[
\int \left( \frac{5}{\sqrt{x}} + 5 \sqrt{x} \right) \, dx = \frac{10}{3} x^{\frac{3}{2}} + 10 \sqrt{x} + C
\][/tex]
where [tex]\(C\)[/tex] is the constant of integration.
