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. We will apply the rules of integration to find the antiderivative of the function.
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
- Identify the Function to Integrate
We need to integrate the function $\frac{x^2}{6}$ with respect to $x$.
- Set Up the Integral
Write the integral in proper form: $$ \int \frac{x^2}{6} , dx $$
- Factor Out the Constant
Since $\frac{1}{6}$ is a constant, we can factor it out of the integral: $$ \frac{1}{6} \int x^2 , dx $$
- Apply the Power Rule of Integration
The power rule states that $\int x^n , dx = \frac{x^{n+1}}{n+1} + C$ where $C$ is the constant of integration. For our case, $n=2$: $$ \int x^2 , dx = \frac{x^{3}}{3} + C $$ So we have: $$ \frac{1}{6} \left( \frac{x^{3}}{3} + C \right) $$
- Combine the Results
Finally, simplifying gives us the result: $$ \frac{x^{3}}{18} + C $$
The integral of the function $\frac{x^2}{6}$ with respect to $x$ is $$ \frac{x^{3}}{18} + C $$.
More Information
The process shown utilized the power rule of integration, which is foundational in calculus. This rule allows us to find the antiderivative of polynomial functions efficiently.
Tips
- Forgetting to include the constant of integration $C$ at the end of the integral.
- Misapplying the power rule (e.g., using $n$ incorrectly) can lead to incorrect results.
