What is the integral of x squared divided by 6?
Understand the Problem
The question is asking for the integral of the function (x^2)/6 with respect to x. This requires applying the rules of integration to find the antiderivative.
Answer
$$ \frac{x^3}{18} + C $$
Answer for screen readers
The integral of the function ( \frac{x^2}{6} ) with respect to ( x ) is
$$ \frac{x^3}{18} + C $$
Steps to Solve
- Identify the Function to Integrate
We need to integrate the function ( f(x) = \frac{x^2}{6} ).
- Set up the Integral
The integral can be set up as follows:
$$ \int \frac{x^2}{6} , dx $$
- Factor out the Constant
Since ( \frac{1}{6} ) is a constant, we can factor it out of the integral:
$$ \int \frac{x^2}{6} , dx = \frac{1}{6} \int x^2 , dx $$
- Integrate the Function
Now we will integrate ( x^2 ) using the power rule of integration:
$$ \int x^n , dx = \frac{x^{n+1}}{n+1} + C $$
Here, ( n = 2 ), so:
$$ \int x^2 , dx = \frac{x^{2+1}}{2+1} + C = \frac{x^3}{3} + C $$
- Combine Results
Now, substitute this result back into the equation:
$$ \int \frac{x^2}{6} , dx = \frac{1}{6} \left( \frac{x^3}{3} + C \right) $$
- Simplify the Expression
Distributing ( \frac{1}{6} ) gives us:
$$ \int \frac{x^2}{6} , dx = \frac{x^3}{18} + C $$
The integral of the function ( \frac{x^2}{6} ) with respect to ( x ) is
$$ \frac{x^3}{18} + C $$
More Information
This result shows the antiderivative of the function. The constant ( C ) represents the constant of integration, which is included because the integral may represent a family of functions.
Tips
- Forgetting to include the constant of integration ( C ).
- Misapplying the power rule, such as getting the exponent wrong or dividing by the incorrect number.
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