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

  1. Identify the Function to Integrate

We need to integrate the function $\frac{x^2}{6}$ with respect to $x$.

  1. Set Up the Integral

Write the integral in proper form: $$ \int \frac{x^2}{6} , dx $$

  1. Factor Out the Constant

Since $\frac{1}{6}$ is a constant, we can factor it out of the integral: $$ \frac{1}{6} \int x^2 , dx $$

  1. 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: $$ \frac{1}{6} \left( \frac{x^{3}}{3} + C \right) $$

  1. 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.
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