What is the integral of x squared over x divided by 6?
Understand the Problem
The question is asking for the integral of the expression x squared divided by 6. We will perform the integration with respect to x.
Answer
$$ \frac{x^3}{18} + C $$
Answer for screen readers
The integral of $\frac{x^2}{6}$ with respect to $x$ is:
$$ \frac{x^3}{18} + C $$
Steps to Solve
- Set up the integral
We need to set up the integral of the function $\frac{x^2}{6}$ with respect to $x$. This can be written as:
$$ \int \frac{x^2}{6} , dx $$
- 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
Now, we will compute the integral of $x^2$. The integral of $x^n$ is given by the formula:
$$ \int x^n , dx = \frac{x^{n+1}}{n+1} + C $$
With $n=2$, we have:
$$ \int x^2 , dx = \frac{x^{2+1}}{2+1} + C = \frac{x^3}{3} + C $$
- Combine the results
Substituting back into our previous step gives:
$$ \frac{1}{6} \left( \frac{x^3}{3} + C \right) $$
- Write the final answer
Simplifying this expression, we get the final result:
$$ \frac{x^3}{18} + C $$
The integral of $\frac{x^2}{6}$ with respect to $x$ is:
$$ \frac{x^3}{18} + C $$
More Information
The result is a polynomial function representing the area under the curve of $\frac{x^2}{6}$. The constant $C$ represents the constant of integration, which is added since we are dealing with indefinite integrals.
Tips
- Forgetting to include the constant of integration $C$ in the final answer.
- Not factoring out the constant before integrating, which can lead to more complex calculations.
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