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

1. 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: $$\int \frac{6}{x} , dx$$

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

3. 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.

4. 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.
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