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. This requires applying the rules of integration to find the antiderivative.
Answer
$$ \frac{x^{3}}{18} + C $$
Answer for screen readers
The final answer is:
$$ \frac{x^{3}}{18} + C $$
Steps to Solve
- Identify the integral to be solved
We want to find 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:
$$ \frac{1}{6} \int x^2 , dx $$
- Find the antiderivative of $x^2$
The antiderivative of $x^2$ is given by the formula $\frac{x^{n+1}}{n+1}$ where $n = 2$ in this case. Therefore, we have:
$$ \int x^2 , dx = \frac{x^{3}}{3} $$
- Combine the results
Now substitute this result back into the equation:
$$ \frac{1}{6} \cdot \frac{x^{3}}{3} = \frac{x^{3}}{18} $$
- Add the constant of integration
When finding an indefinite integral, we must also add a constant of integration, $C$:
$$ \int \frac{x^2}{6} , dx = \frac{x^{3}}{18} + C $$
The final answer is:
$$ \frac{x^{3}}{18} + C $$
More Information
The constant of integration, $C$, represents all possible antiderivatives of the function and is essential in indefinite integrals. This is because there are infinite functions that differ by a constant that have the same derivative.
Tips
- Forgetting to include the constant of integration, $C$, in the final answer is a common mistake.
- Misapplying the power rule when finding the antiderivative can lead to incorrect results. Always ensure you increase the exponent by 1 and divide by the new exponent.
