The antiderivative (or indefinite integral) of the function $\frac{4}{x}$ can be found using the basic formula for integrating functions of the form $\frac{1}{x}$.

The integral of $\frac{1}{x}$ is $\ln|x|$. So, for $\frac{4}{x}$, we can factor out the constant 4 and integrate $\frac{1}{x}$ as follows:

$\int \frac{4}{x} \, dx = 4 \int \frac{1}{x} \, dx = 4 \ln|x| + C$

where $C$ is the constant of integration.

Final Answer:

$\int \frac{4}{x} \, dx = 4 \ln|x| + C$

Would you like more details on this, or do you have any other questions? Here are some related questions you might find interesting:

1. How does the natural logarithm function behave graphically?
2. What is the integral of other rational functions like $\frac{1}{x^2}$?
3. Can the antiderivative formula for $\frac{1}{x}$ be extended to complex numbers?
4. What is the connection between integration and areas under curves?
5. How do you compute the antiderivative of a function with multiple terms, like $\frac{4}{x} + x^2$?

Tip: Always remember to include the absolute value inside the logarithm when dealing with $\frac{1}{x}$, since the natural logarithm is only defined for positive numbers, and the absolute value accounts for both positive and negative values of $x$.