The problem involves solving indefinite integrals for three given functions. Let’s calculate the indefinite integral for each:

1. $f(x) = \int (3x^2 + 2x - 1) \, dx$

To integrate, we apply the power rule: $\int x^n \, dx = \frac{x^{n+1}}{n+1}, \quad n \neq -1.$ Thus: $\int (3x^2 + 2x - 1) \, dx = \int 3x^2 \, dx + \int 2x \, dx - \int 1 \, dx$ $= 3 \cdot \frac{x^{3}}{3} + 2 \cdot \frac{x^2}{2} - x + C.$ Simplify: $f(x) = x^3 + x^2 - x + C.$

2. $f(x) = \int (4x^3 - 5x^2 + x) \, dx$

Using the same power rule: $\int (4x^3 - 5x^2 + x) \, dx = \int 4x^3 \, dx - \int 5x^2 \, dx + \int x \, dx$ $= 4 \cdot \frac{x^4}{4} - 5 \cdot \frac{x^3}{3} + \frac{x^2}{2} + C.$ Simplify: $f(x) = x^4 - \frac{5x^3}{3} + \frac{x^2}{2} + C.$

3. $f(x) = \int (2\sin(x) + \cos(x)) \, dx$

For trigonometric integrals: $\int \sin(x) \, dx = -\cos(x), \quad \int \cos(x) \, dx = \sin(x).$ Thus: $\int (2\sin(x) + \cos(x)) \, dx = 2 \int \sin(x) \, dx + \int \cos(x) \, dx$ $= 2(-\cos(x)) + \sin(x) + C.$ Simplify: $f(x) = -2\cos(x) + \sin(x) + C.$

Final Results:

1. $f(x) = x^3 + x^2 - x + C$,
2. $f(x) = x^4 - \frac{5x^3}{3} + \frac{x^2}{2} + C$,
3. $f(x) = -2\cos(x) + \sin(x) + C$.

Would you like step-by-step explanations or details on how I applied the integration rules?

Related Questions:

1. What are the basic integration rules and how are they applied?
2. How do you handle integrals involving exponential or logarithmic functions?
3. Can integration by substitution be used for trigonometric functions?
4. What are definite integrals, and how are they calculated?
5. How is the constant of integration $C$ interpreted in applications?

Tip: Always check your results by differentiating your answer to verify if it matches the original function.