What is the integral of x squared divided by 6?

Steps to Solve

1. Identify the Function to Integrate

We need to integrate the function ( f(x) = \frac{x^2}{6} ).

2. Set up the Integral

The integral can be set up as follows:

$$ \int \frac{x^2}{6} , dx $$

3. Factor out the Constant

Since ( \frac{1}{6} ) is a constant, we can factor it out of the integral:

$$ \int \frac{x^2}{6} , dx = \frac{1}{6} \int x^2 , dx $$

4. 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:

$$ \int x^2 , dx = \frac{x^{2+1}}{2+1} + C = \frac{x^3}{3} + C $$

5. Combine Results

Now, substitute this result back into the equation:

$$ \int \frac{x^2}{6} , dx = \frac{1}{6} \left( \frac{x^3}{3} + C \right) $$

6. Simplify the Expression

Distributing ( \frac{1}{6} ) gives us:

$$ \int \frac{x^2}{6} , dx = \frac{x^3}{18} + C $$