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. To solve this, we will apply the basic rules of integration to find the antiderivative of the given function.
Answer
$$ \frac{x^3}{18} + C $$
Answer for screen readers
The integral of $\frac{x^2}{6}$ with respect to x is: $$ \frac{x^3}{18} + C $$
Steps to Solve
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Set up the integral We are asked to find the integral of the function $\frac{x^2}{6}$ with respect to x. This can be written as: $$ \int \frac{x^2}{6} , dx $$
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Factor out constants Since $\frac{1}{6}$ is a constant, we can factor it out of the integral: $$ \frac{1}{6} \int x^2 , dx $$
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Use the power rule of integration Now we apply the power rule of integration which states that: $$ \int x^n , dx = \frac{x^{n+1}}{n+1} + C $$ For our case, $n = 2$. Therefore: $$ \int x^2 , dx = \frac{x^{2+1}}{2+1} + C = \frac{x^3}{3} + C $$
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Combine the results Substituting back into our factored integral, we have: $$ \frac{1}{6} \left( \frac{x^3}{3} + C \right) = \frac{x^3}{18} + C $$
The integral of $\frac{x^2}{6}$ with respect to x is: $$ \frac{x^3}{18} + C $$
More Information
The integral we found represents the area under the curve of the function $\frac{x^2}{6}$ over the x-axis. The constant $C$ represents the constant of integration, which accounts for the family of antiderivatives.
Tips
- Forgetting to include the constant of integration $C$ is a frequent error. Always remember to add $C$ when solving indefinite integrals.
