What is the integral of x squared over x divided by 6?

Understand the Problem

The question asks for the integral of a mathematical expression involving x squared divided by 6. The expression can be simplified and integrated accordingly.

Answer

$$ \frac{x^{3}}{18} + C $$
Answer for screen readers

$$ \frac{x^{3}}{18} + C $$

Steps to Solve

  1. Identify the Integral to Solve

We need to find the integral of the expression $\frac{x^2}{6}$ with respect to $x$.

  1. Set Up the Integral

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

  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. Integrate the Function

Now, we need to calculate the integral of $x^2$. Recall the power rule of integration, which states that: $$ \int x^n , dx = \frac{x^{n+1}}{n+1} + C $$

For $n=2$, this gives us: $$ \int x^2 , dx = \frac{x^{3}}{3} + C $$

  1. Combine the Results

Now, substitute the result back into the equation: $$ \frac{1}{6} \left( \frac{x^{3}}{3} + C \right) $$

Which can be simplified to: $$ \frac{x^{3}}{18} + \frac{C}{6} $$

  1. Write the Final Result

To express the integral result, we typically denote any constant of integration simply as $C$: $$ \frac{x^{3}}{18} + C $$

$$ \frac{x^{3}}{18} + C $$

More Information

The integral $\frac{x^{3}}{18} + C$ represents the area under the curve of the function $\frac{x^2}{6}$ on the x-axis, plus a constant of integration $C$, which can represent any vertical shift of the function.

Tips

  • Forgetting to factor out the constant $\frac{1}{6}$ before integrating.
  • Misapplying the power rule of integration, particularly the increment in the exponent and the division by the new exponent.
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