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