Let's solve each of these integrals step-by-step.

Problem 1:

$\int (X - 4) \, dx$ Separate the terms and integrate: $= \int X \, dx - \int 4 \, dx$ The integrals are: $= \frac{X^2}{2} - 4X + C$

Answer: $\frac{X^2}{2} - 4X + C$

Problem 2:

$\int (3X + 4) \, dx$ Separate the terms and integrate: $= \int 3X \, dx + \int 4 \, dx$ The integrals are: $= \frac{3X^2}{2} + 4X + C$

Answer: $\frac{3X^2}{2} + 4X + C$

Problem 3:

$\int (X + 8) \, dx$ Separate the terms and integrate: $= \int X \, dx + \int 8 \, dx$ The integrals are: $= \frac{X^2}{2} + 8X + C$

Answer: $\frac{X^2}{2} + 8X + C$

Problem 4:

$\int 3X(4X^2 + 2X) \, dx$ First, expand the terms inside the integral: $= \int 3X(4X^2) \, dx + \int 3X(2X) \, dx$ $= \int 12X^3 \, dx + \int 6X^2 \, dx$ Now, integrate each term: $= \frac{12X^4}{4} + \frac{6X^3}{3} + C$ Simplify: $= 3X^4 + 2X^3 + C$

Answer: $3X^4 + 2X^3 + C$

Would you like more details or have any questions?

Related Questions

1. How would you handle integrals involving trigonometric functions?
2. What are some common techniques for integrating products of functions?
3. How do you apply integration to solve area under curves?
4. What is the role of the constant $C$ in indefinite integrals?
5. How can substitution help in simplifying integrals?

Tip

When integrating, always check if the expression can be simplified before applying the integration rules.