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

  1. Set up the integral We want to find the integral of the function $\frac{x^2}{6}$. This can be written as: $$ \int \frac{x^2}{6} , dx $$

  2. Factor out the constant When integrating, we can factor out constants from the integral. Here, we can factor out $\frac{1}{6}$: $$ \frac{1}{6} \int x^2 , dx $$

  3. 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: $$ \int x^2 , dx = \frac{x^{3}}{3} + C $$

  4. Combine the results Now, we substitute back into our factored integral: $$ \frac{1}{6} \left(\frac{x^{3}}{3} + C\right) $$

  5. 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.
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