What is the integral of the function (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 using integration rules.
Answer
$$ \frac{x^3}{18} + C $$
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
Write the integral in mathematical notation. We need to find the integral of the function $\frac{x^2}{6}$ with respect to $x$.
$$ \int \frac{x^2}{6} , dx $$
- Factor out 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
Next, we need to find the antiderivative of $x^2$. The general formula for the antiderivative is:
$$ \int x^n , dx = \frac{x^{n+1}}{n+1} + C \text{, where } n \neq -1 $$
For $n = 2$, this becomes:
$$ \int x^2 , dx = \frac{x^{2+1}}{2+1} + C = \frac{x^3}{3} + C $$
- Combine results
Now, substitute back into the equation along with the constant we factored out:
$$ \frac{1}{6} \left( \frac{x^3}{3} + C \right) $$
This simplifies to:
$$ \frac{x^3}{18} + C $$
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 area under the curve of the function $\frac{x^2}{6}$. The constant $C$ represents any constant of integration that arises from indefinite integrals.
Tips
- Forgetting the constant C: Always remember to include the constant of integration when calculating indefinite integrals.
- Improperly applying integration rules: Make sure to apply the power rule correctly and remember that the term must be raised to the next higher power and divided by that power.
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