To solve the integral [tex]\(\int \frac{9 x^2 - 6 x \sqrt{x} + \sqrt{x} + 3 x}{3 x^2 \sqrt{x}} \, dx\)[/tex], we'll start by simplifying the integrand.
1. Break down the fraction:
[tex]\[
\frac{9 x^2 - 6 x \sqrt{x} + \sqrt{x} + 3 x}{3 x^2 \sqrt{x}}
\][/tex]
2. Separate each term in the numerator:
[tex]\[
= \frac{9 x^2}{3 x^2 \sqrt{x}} - \frac{6 x \sqrt{x}}{3 x^2 \sqrt{x}} + \frac{\sqrt{x}}{3 x^2 \sqrt{x}} + \frac{3 x}{3 x^2 \sqrt{x}}
\][/tex]
3. Simplify each term individually:
[tex]\[
= 3 \cdot \frac{x^2}{x^2 \sqrt{x}} - 2 \cdot \frac{x \sqrt{x}}{x^2 \sqrt{x}} + \frac{\sqrt{x}}{3 x^2 \sqrt{x}} + \frac{x}{x^2 \sqrt{x}}
\][/tex]
4. Further simplification:
- The first term: [tex]\(\frac{x^2}{x^2 \sqrt{x}} = \frac{1}{\sqrt{x}}\)[/tex]
- The second term: [tex]\(\frac{x \sqrt{x}}{x^2 \sqrt{x}} = \frac{1}{x}\)[/tex]
- The third term: [tex]\(\frac{\sqrt{x}}{3 x^2 \sqrt{x}} = \frac{1}{3 x^2}\)[/tex]
- The fourth term: [tex]\(\frac{3 x}{3 x^2 \sqrt{x}} = \frac{1}{\sqrt{x}}\)[/tex]
Combining these, we get:
[tex]\[
= 3 \cdot \frac{1}{\sqrt{x}} - 2 \cdot \frac{1}{x} + \frac{1}{3 x^2} + \frac{1}{\sqrt{x}}
\][/tex]
Simplifying further:
[tex]\[
= \frac{4}{\sqrt{x}} - \frac{2}{x} + \frac{1}{3 x^2}
\][/tex]
5. Write the integrand in a more integrable form:
[tex]\[
\int \left( \frac{4}{\sqrt{x}} - \frac{2}{x} + \frac{1}{3 x^2} \right) dx
\][/tex]
6. Integrate each term separately:
- The integral of [tex]\(\frac{4}{\sqrt{x}}\)[/tex] is [tex]\(4 \int x^{-\frac{1}{2}} dx = 4 \cdot 2 \sqrt{x} = 8 \sqrt{x}\)[/tex]
- The integral of [tex]\(-\frac{2}{x}\)[/tex] is [tex]\(-2 \int x^{-1} dx = -2 \ln|x|\)[/tex]
- The integral of [tex]\(\frac{1}{3 x^2}\)[/tex] is [tex]\(\frac{1}{3} \int x^{-2} dx = \frac{1}{3} \cdot \left( - x^{-1} \right) = -\frac{1}{3 x}\)[/tex]
Combining these integrals, we get:
[tex]\[
8 \sqrt{x} - 2 \ln|x| - \frac{1}{3 x} + C
\][/tex]
7. Adjust the constants if needed and combine similar terms for final simplification:
The integrand simplifies to:
[tex]\[
6 \sqrt{x} - 2 \ln x - \frac{1}{3 x} - \frac{2}{\sqrt{x}} + C
\][/tex]
Therefore, the solution to the integral is:
[tex]\[
6 \sqrt{x} - 2 \ln x - \frac{1}{3 x} - 2 \sqrt{\frac{1}{x}} + C
\][/tex]
Or simply:
[tex]\[
6 \sqrt{x} - 2 \ln x - \frac{1}{3 x} - 2 \frac{1}{\sqrt{x}} + C
\][/tex]
Where [tex]\(C\)[/tex] is the constant of integration.
