What is the integral of x squared over x divided by 6?
Understand the Problem
The question is asking for the integral of the function ( \frac{x^2}{x} ) divided by 6, which simplifies to ( \frac{x}{6} ). The integral can be calculated using basic integral rules.
Answer
$$ \int \frac{x}{6} \, dx = \frac{x^2}{12} + C $$
Answer for screen readers
The final answer for the integral is $$ \int \frac{x}{6} , dx = \frac{x^2}{12} + C $$
Steps to Solve
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Set up the integral We start with the function given, which has been simplified to ( \frac{x}{6} ). We need to find the integral $$ \int \frac{x}{6} , dx $$
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Factor out the constant Since ( \frac{1}{6} ) is a constant, we can factor it out of the integral: $$ \int \frac{x}{6} , dx = \frac{1}{6} \int x , dx $$
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Integrate the function Now, we apply the power rule for integration. The integral of ( x ) is ( \frac{x^2}{2} ): $$ \int x , dx = \frac{x^2}{2} $$
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Combine results Putting it all together, we have: $$ \frac{1}{6} \int x , dx = \frac{1}{6} \cdot \frac{x^2}{2} = \frac{x^2}{12} $$
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Add the constant of integration Don't forget to add the constant of integration ( C ): $$ \int \frac{x}{6} , dx = \frac{x^2}{12} + C $$
The final answer for the integral is $$ \int \frac{x}{6} , dx = \frac{x^2}{12} + C $$
More Information
This integral represents the area under the curve of the function ( \frac{x}{6} ) with respect to ( x ). Adding the constant ( C ) accounts for any vertical shifts of the antiderivative.
Tips
- Forgetting the constant of integration: Always remember to add ( C ) at the end of your integration to represent all possible antiderivatives.
- Incorrectly applying the power rule: Ensure proper application of the power rule when dealing with polynomial functions.
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