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

  1. Identify the Function to Integrate

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

  1. Set up the Integral

The integral can be set up as follows:

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

  1. Factor out the Constant

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

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

  1. 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:

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

  1. Combine Results

Now, substitute this result back into the equation:

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

  1. 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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