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

Understand the Problem

The question is asking for the integral of the function (x^2)/6 with respect to x, which involves finding the antiderivative of the given function using integration rules.

Answer

$$ \frac{x^{3}}{18} + C $$
Answer for screen readers

The final answer to the integral is:

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

Steps to Solve

  1. Identify the function to integrate

We want to find the integral of the function $\frac{x^2}{6}$.

  1. Set up the integral

We can set up the integral as follows:

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

  1. Simplify the integral

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

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

  1. Find the antiderivative of the function

The antiderivative of $x^n$ is given by the formula $\frac{x^{n+1}}{n+1} + C$, where $C$ is the constant of integration. For our case with $n=2$:

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

  1. Combine and write the final result

Now, we substitute back into our integral:

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

This simplifies to:

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

The final answer to the integral is:

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

More Information

This integral represents the area under the curve of the function $\frac{x^2}{6}$ for any interval along the x-axis. The constant $C$ represents any constant of integration since indefinite integrals can vary by a constant.

Tips

  • Mistaking the power rule for integration: Remember to add 1 to the exponent and divide by the new exponent when finding the antiderivative.
  • Forgetting to add the constant of integration $C$, which is crucial in indefinite integrals.

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