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
- 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:
$$ \int \frac{x^2}{6} , dx $$
- Factor out the constant
Since $\frac{1}{6}$ is a constant, we can factor it out of the integral:
$$ \frac{1}{6} \int x^2 , dx $$
- 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:
$$ \int x^2 , dx = \frac{x^{2+1}}{2+1} + C = \frac{x^3}{3} + C $$
- 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.
