What is the integral of x squared divided by 6?
Understand the Problem
The question is asking for the integral of the function x squared divided by 6. To solve this, we can simplify the expression and then apply the rules of integration to find the antiderivative.
Answer
The integral of $\frac{x^2}{6}$ is $\frac{x^{3}}{18} + C$.
Answer for screen readers
The integral of $\frac{x^2}{6}$ is $\frac{x^{3}}{18} + C$.
Steps to Solve
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Set up the integral We want to find the integral of the function $\frac{x^2}{6}$. This can be written as: $$ \int \frac{x^2}{6} , dx $$
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Factor out the constant When integrating, we can factor out constants from the integral. Here, we can factor out $\frac{1}{6}$: $$ \frac{1}{6} \int x^2 , dx $$
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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 $n = 2$, we have: $$ \int x^2 , dx = \frac{x^{3}}{3} + C $$
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Combine the results Now, we substitute back into our factored integral: $$ \frac{1}{6} \left(\frac{x^{3}}{3} + C\right) $$
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Simplify the expression Multiply the constants: $$ \frac{x^{3}}{18} + \frac{C}{6} $$
Thus the final result is: $$ \frac{x^{3}}{18} + C $$
The integral of $\frac{x^2}{6}$ is $\frac{x^{3}}{18} + C$.
More Information
The constant $C$ represents the constant of integration, which is added because integration is an indefinite process. This means that there are infinitely many antiderivatives for any given function.
Tips
- Forgetting to add the constant of integration $C$ after finding the antiderivative.
- Not factoring out constants before integrating, which can make calculations more complicated.
