What is the integral of (x^2)/6 with respect to x?

Understand the Problem

The question is asking us to find the integral of the function (x^2)/6 with respect to x. To solve this, we will apply the rules of integration to determine the antiderivative of the given function.

Answer

The integral of $\frac{x^2}{6}$ with respect to $x$ is $\frac{x^3}{18} + C$.
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. Set up the integral

We want to find the integral of the function $\frac{x^2}{6}$ with respect to $x$. We set it 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:

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

  1. Integrate the function

Next, we apply the power rule of integration. The power rule states that:

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

For our function, $n=2$. So:

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

  1. Combine the results

Now, we bring everything together:

$$ \frac{1}{6} \left( \frac{x^3}{3} + C \right) $$

This simplifies to:

$$ \frac{x^3}{18} + C' $$

Where $C' = \frac{C}{6}$ is also a constant.

The integral of the function $\frac{x^2}{6}$ with respect to $x$ is:

$$ \frac{x^3}{18} + C $$

More Information

The result represents the family of functions that are the antiderivative of $\frac{x^2}{6}$. The constant $C$ represents an arbitrary constant that can be determined if initial conditions are given.

Tips

  • Forgetting to include the constant of integration $C$ is a common mistake. Always remember to add this constant when finding indefinite integrals.
  • Misapplying the power rule can lead to incorrect integration results.

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