Let's solve the given integral:

The integral is:

$\int \frac{x^2 - \sqrt{x} + 10}{x} \, dx$

Step 1: Simplify the expression

Split the fraction into separate terms: $\frac{x^2}{x} - \frac{\sqrt{x}}{x} + \frac{10}{x} = x - x^{-1/2} + 10x^{-1}$

So, the integral becomes: $\int \left( x - x^{-1/2} + 10x^{-1} \right) \, dx$

Step 2: Integrate term by term

1. For $\int x \, dx$: $\frac{x^2}{2}$

2. For $\int x^{-1/2} \, dx$: $\frac{x^{1/2}}{1/2} = 2x^{1/2}$

3. For $\int 10x^{-1} \, dx$: $10 \ln|x|$

Step 3: Combine results

The integral is: $\frac{x^2}{2} - 2x^{1/2} + 10 \ln|x| + C$

Final Answer:

$\int \frac{x^2 - \sqrt{x} + 10}{x} \, dx = \frac{x^2}{2} - 2\sqrt{x} + 10 \ln|x| + C$

Let me know if you'd like further clarification or have additional questions.

Here are 5 related questions to expand your understanding:

1. How do you handle integrals with multiple terms in the numerator?
2. What is the general rule for integrating $x^n$?
3. Why do we include the constant of integration $C$?
4. How is the natural logarithm $\ln|x|$ derived during integration?
5. Can substitution be used to simplify this integral?

Tip: Always simplify fractions before integrating—it makes solving much easier!