To find the integral of the given function $f(x) = \frac{x - 3x^2}{\sqrt{x}}$, we need to simplify the integrand first.

Given: $f(x) = \frac{x - 3x^2}{\sqrt{x}}$

We can break it down into two separate fractions: $f(x) = \frac{x}{\sqrt{x}} - \frac{3x^2}{\sqrt{x}}$

Simplify each term: $\frac{x}{\sqrt{x}} = x^{1 - \frac{1}{2}} = x^{\frac{1}{2}}$ $\frac{3x^2}{\sqrt{x}} = 3x^{2 - \frac{1}{2}} = 3x^{\frac{3}{2}}$

So, the function $f(x)$ becomes: $f(x) = x^{\frac{1}{2}} - 3x^{\frac{3}{2}}$

Now, we integrate $f(x)$: $\int f(x) \, dx = \int \left( x^{\frac{1}{2}} - 3x^{\frac{3}{2}} \right) dx$

Using the power rule for integration: $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$

Apply the rule to each term: $\int x^{\frac{1}{2}} \, dx = \int x^{\frac{1}{2}} \, dx = \frac{x^{\frac{1}{2} + 1}}{\frac{1}{2} + 1} = \frac{x^{\frac{3}{2}}}{\frac{3}{2}} = \frac{2}{3} x^{\frac{3}{2}}$

$\int 3x^{\frac{3}{2}} \, dx = 3 \int x^{\frac{3}{2}} \, dx = 3 \cdot \frac{x^{\frac{3}{2} + 1}}{\frac{3}{2} + 1} = 3 \cdot \frac{x^{\frac{5}{2}}}{\frac{5}{2}} = 3 \cdot \frac{2}{5} x^{\frac{5}{2}} = \frac{6}{5} x^{\frac{5}{2}}$

Combine the integrals: $\int \left( x^{\frac{1}{2}} - 3x^{\frac{3}{2}} \right) dx = \frac{2}{3} x^{\frac{3}{2}} - \frac{6}{5} x^{\frac{5}{2}} + C$

Thus, the integral of $f(x)$ is: $\int f(x) \, dx = \frac{2}{3} x^{\frac{3}{2}} - \frac{6}{5} x^{\frac{5}{2}} + C$

Would you like more details or have any questions about this solution?

Here are some related questions you might find useful:

1. How do you apply the power rule for integration to other functions?
2. What are the steps to simplify a complex fraction before integrating?
3. How do you determine the appropriate bounds for a definite integral?
4. Can you provide an example of integrating a trigonometric function?
5. How do you handle integration by parts for more complex integrands?
6. What is the substitution method, and when is it useful in integration?
7. How do you integrate a rational function using partial fractions?
8. What are some common mistakes to avoid when integrating by hand?

Tip: When integrating functions, always look for ways to simplify the integrand before applying integration rules. This can make the process much easier and less error-prone.