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
- Identify the Integral to Solve
We need to find the integral of the expression $\frac{x^2}{6}$ with respect to $x$.
- Set Up the Integral
Write the integral as: $$ \int \frac{x^2}{6} , dx $$
- Factor Out the Constant
Since $\frac{1}{6}$ is a constant, we can factor it out of the integral: $$ \frac{1}{6} \int x^2 , dx $$
- 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: $$ \int x^2 , dx = \frac{x^{3}}{3} + C $$
- 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: $$ \frac{x^{3}}{18} + \frac{C}{6} $$
- 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.
