What is the integral of x squared over 6?

Understand the Problem

The question is asking to find the integral of the function x squared divided by 6. This involves applying the basic rules of integration.

Answer

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

Steps to Solve

Identify the function to integrate

The function we want to integrate is given as $\frac{x^2}{6}$.

Use the rule for integrating a power function

The integral of $x^n$ is given by the formula:

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

In this case, we have $n = 2$, so we rewrite the integral:

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

Calculate the integral of $x^2$

Using the rule from the previous step, we compute:

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

Combine the results

Now we substitute back into our previous equation:

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

Simplify the expression

This simplifies to:

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

The integral of the function is $$ \frac{x^3}{18} + C $$

More Information

In the solution, we found the integral of a simple polynomial function. Integrals are a fundamental concept in calculus that help determine areas under curves. This specific integral can be interpreted as the area under the curve of the function $\frac{x^2}{6}$.

Tips

Forgetting the constant of integration ($C$): It's important to remember that after integrating, you should always include the constant because the integral represents a family of functions.
Mistaking the power rule: Ensure you are correctly applying the power rule to the correct exponent.

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