What is the integral of x squared over 6?
Understand the Problem
The question is asking for the integral of the function x squared divided by 6 with respect to x. This is a calculus question that involves finding the antiderivative, which can be solved using basic integration rules.
Answer
The integral is $ \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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Identify the Integral We need to evaluate the integral of the function $\frac{x^2}{6}$ with respect to $x$. The expression can be rewritten as: $$ \int \frac{x^2}{6} , dx $$
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Factor Out the Constant Since $\frac{1}{6}$ is a constant, we can factor it out of the integral: $$ \frac{1}{6} \int x^2 , dx $$
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Integrate the Function Next, we apply the power rule of integration. The power rule states that: $$ \int x^n , dx = \frac{x^{n+1}}{n+1} + C $$ For our case where $n=2$, we get: $$ \int x^2 , dx = \frac{x^{3}}{3} + C $$
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Combine Results Now substitute back into the equation: $$ \frac{1}{6} \left( \frac{x^3}{3} + C \right) $$
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Simplify the Expression Distributing the factor of $\frac{1}{6}$, we have: $$ \frac{x^3}{18} + \frac{C}{6} $$
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Final Formulation The constant $\frac{C}{6}$ can simply be represented as a constant $C'$. Therefore, the final result can be presented as: $$ \frac{x^3}{18} + C' $$
The integral of $\frac{x^2}{6}$ with respect to $x$ is: $$ \frac{x^3}{18} + C $$
More Information
When you find the integral, you're essentially determining the area under the curve of the function. The constant $C$ represents the family of all antiderivatives, as integrating an expression can yield multiple forms differing by a constant.
Tips
One common mistake is forgetting to apply the power rule correctly or miscalculating the exponent when integrating. Always remember to add 1 to the exponent and divide by the new exponent.
