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 of the given function.
Answer
The integral is $\frac{x^3}{18} + C$.
Answer for screen readers
The integral of the function $\frac{x^2}{6}$ with respect to $x$ is:
$$ \int \frac{x^2}{6} , dx = \frac{x^3}{18} + C $$
Steps to Solve
- Identify the function to integrate
We need to integrate the function $f(x) = \frac{x^2}{6}$, which is the same as $\frac{1}{6} x^2$.
- Apply the power rule of integration
The power rule states that the integral of $x^n$ is $\frac{x^{n+1}}{n+1}$, where $n \neq -1$.
Here, we can say: $$ \int \frac{x^2}{6} , dx = \frac{1}{6} \int x^2 , dx $$
- Calculate the integral using the power rule
Now, apply the power rule for $n = 2$: $$ \int x^2 , dx = \frac{x^{2+1}}{2+1} = \frac{x^3}{3} $$
Therefore: $$ \frac{1}{6} \int x^2 , dx = \frac{1}{6} \cdot \frac{x^3}{3} = \frac{x^3}{18} $$
- Add the constant of integration
Since this is an indefinite integral, we need to add a constant $C$ to the result: $$ \int \frac{x^2}{6} , dx = \frac{x^3}{18} + C $$
The integral of the function $\frac{x^2}{6}$ with respect to $x$ is:
$$ \int \frac{x^2}{6} , dx = \frac{x^3}{18} + C $$
More Information
This integral represents the area under the curve of the function $\frac{x^2}{6}$ from a specific point to another point along the x-axis. The constant $C$ accounts for all possible horizontal shifts of the antiderivative.
Tips
- Forgetting to add the constant of integration ( C ). Always remember to include it in indefinite integrals!
- Not applying the power rule correctly, especially confusing the signs or the coefficients.
AI-generated content may contain errors. Please verify critical information
