The antiderivative of $4x$ can be found by applying the power rule for integration. The power rule states:

$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$

where $C$ is the constant of integration, and $n \neq -1$.

For the function $4x$, we have $n = 1$, so:

$\int 4x \, dx = 4 \cdot \frac{x^{1+1}}{1+1} + C = 4 \cdot \frac{x^2}{2} + C = 2x^2 + C$

Thus, the antiderivative of $4x$ is:

$2x^2 + C$

Let me know if you'd like more details or further explanations!

Related Questions:

1. How would you integrate a polynomial like $3x^3 + 2x^2 + x$?
2. What is the antiderivative of $5x^4$?
3. Can you explain the constant of integration in an antiderivative?
4. How do you use the power rule for higher powers of $x$?
5. How do definite integrals differ from indefinite integrals?

Tip:

When integrating, always remember to add the constant of integration $C$ unless you are given specific bounds for a definite integral.