What is the integral of (x^2)/6 with respect to x?
Understand the Problem
The question is asking us to find the integral of the function (x^2)/6 with respect to x, which involves applying integration rules to determine the antiderivative.
Answer
$$ \int \frac{x^2}{6} \, dx = \frac{x^{3}}{18} + C $$
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 integral to solve We need to calculate the integral of the function $\frac{x^2}{6}$. This can be expressed as: $$ \int \frac{x^2}{6} , dx $$
-
Factor out constants from the integral Since $\frac{1}{6}$ is a constant, we can factor it out of the integral: $$ \int \frac{x^2}{6} , dx = \frac{1}{6} \int x^2 , dx $$
-
Apply the power rule of integration Using the power rule, which states that $\int x^n , dx = \frac{x^{n+1}}{n+1} + C$, with $n = 2$: $$ \int x^2 , dx = \frac{x^{3}}{3} + C $$
-
Combine results Now plug the result of the integral back in with the constant: $$ \frac{1}{6} \left( \frac{x^{3}}{3} + C \right) = \frac{x^{3}}{18} + \frac{C}{6} $$
-
Final expression for the integral The final expression for the integral becomes: $$ \int \frac{x^2}{6} , dx = \frac{x^{3}}{18} + C $$
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 antiderivative of the function $\frac{x^2}{6}$. The constant $C$ is important as it accounts for any constant value that could be added to the antiderivative, which does not affect its derivative.
Tips
- Forgetting to include the constant of integration $C$ is a common mistake. Always remember that an indefinite integral should include this constant.
AI-generated content may contain errors. Please verify critical information
