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
- Identify the function to integrate
We want to find the integral of the function $\frac{x^2}{6}$.
- Set up the integral
We can set up the integral as follows:
$$ \int \frac{x^2}{6} , dx $$
- Simplify the integral
Since $\frac{1}{6}$ is a constant, we can factor it out of the integral:
$$ \frac{1}{6} \int x^2 , dx $$
- 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 $n=2$:
$$ \int x^2 , dx = \frac{x^{3}}{3} + C $$
- 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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