What is the antiderivative of 6/x?
Understand the Problem
The question is asking for the antiderivative (or indefinite integral) of the function 6/x. This involves finding a function whose derivative is equal to 6/x, which can be solved using integration techniques.
Answer
The antiderivative of $ \frac{6}{x} $ is $ 6 \ln x + C $.
Answer for screen readers
The antiderivative of $ \frac{6}{x} $ is $ 6 \ln x + C $.
Steps to Solve

Identify the Integral to Solve We need to solve the integral of the function $ \frac{6}{x} $ with respect to $ x $. This can be rewritten as: $$ \int \frac{6}{x} , dx $$

Factor Out the Constant Since integration allows us to factor out constants, we can take the number 6 outside of the integral: $$ 6 \int \frac{1}{x} , dx $$

Integrate the Function The integral of $ \frac{1}{x} $ is a standard result. We know that: $$ \int \frac{1}{x} , dx = \ln x + C $$ where $ C $ is the constant of integration.

Combine the Results Now we can combine our results from the previous steps: $$ 6 \int \frac{1}{x} , dx = 6 (\ln x + C) $$ This simplifies to: $$ 6 \ln x + C $$
The antiderivative of $ \frac{6}{x} $ is $ 6 \ln x + C $.
More Information
The result $ 6 \ln x + C $ represents a family of functions whose derivative gives $ \frac{6}{x} $. The $ C $ is an arbitrary constant representing all possible vertical shifts of this function.
Tips
 Forgetting to include the constant of integration $ C $, which is essential in indefinite integrals.
 Not recognizing that the integral of $ \frac{1}{x} $ is not simply $ x $, but rather a logarithmic function.