Let's go through each of the integrals one by one:

1. $\int (x - 2)^2 \, dx$
   Expand the square and integrate: $(x - 2)^2 = x^2 - 4x + 4$ Now integrate term by term: $\int (x^2 - 4x + 4) \, dx = \frac{x^3}{3} - 2x^2 + 4x + C$ So, the solution is: $\frac{x^3}{3} - 2x^2 + 4x + C$

2. $\int \frac{2x^2 + x + 1}{x^3} \, dx$
   Rewrite the integrand as separate terms: $\frac{2x^2 + x + 1}{x^3} = \frac{2x^2}{x^3} + \frac{x}{x^3} + \frac{1}{x^3} = \frac{2}{x} + \frac{1}{x^2} + \frac{1}{x^3}$ Now integrate each term: $\int \frac{2}{x} \, dx = 2\ln|x|, \quad \int \frac{1}{x^2} \, dx = -\frac{1}{x}, \quad \int \frac{1}{x^3} \, dx = -\frac{1}{2x^2}$ So, the solution is: $2\ln|x| - \frac{1}{x} - \frac{1}{2x^2} + C$

3. $\int \frac{1 + \sqrt{x}}{x} \, dx$
   Rewrite the integrand: $\frac{1 + \sqrt{x}}{x} = \frac{1}{x} + \frac{\sqrt{x}}{x} = \frac{1}{x} + x^{-\frac{1}{2}}$ Now integrate each term: $\int \frac{1}{x} \, dx = \ln|x|, \quad \int x^{-\frac{1}{2}} \, dx = 2\sqrt{x}$ So, the solution is: $\ln|x| + 2\sqrt{x} + C$

4. $\int 3(x - 2)^9 \, dx$
   Use substitution or direct integration: Let $u = x - 2$, then $du = dx$. The integral becomes: $3\int u^9 \, du = 3 \cdot \frac{u^{10}}{10} = \frac{3(x - 2)^{10}}{10} + C$

5. $\int x(5x^2 + 2)^9 \, dx$
   Use substitution: Let $u = 5x^2 + 2$, then $du = 10x \, dx$. The integral becomes: $\frac{1}{10}\int u^9 \, du = \frac{1}{10} \cdot \frac{u^{10}}{10} = \frac{(5x^2 + 2)^{10}}{100} + C$

Would you like more details on any of these integrals?

Here are 5 related questions to expand on these topics:

1. How would you approach an integral involving trigonometric functions like $\int \sin^2(x) \, dx$?
2. What is the method of substitution, and how is it used in integration?
3. Can you apply integration by parts to solve any of the problems above?
4. How do you evaluate definite integrals for these examples?
5. How does the chain rule apply in differentiation, and how is it linked to the substitution method in integration?

Tip: Always simplify the integrand as much as possible before attempting to integrate, especially when dealing with powers or fractions.