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. This involves integrating the function to find its antiderivative, which is a fundamental concept in calculus.
Answer
$$ \frac{x^3}{18} + C $$
Answer for screen readers
The final answer for the integral is: $$ \frac{x^3}{18} + C $$
Steps to Solve
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Set Up the Integral We start by writing down the integral we want to solve: $$ \int \frac{x^2}{6} , dx $$
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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 Power Function Next, we apply the power rule of integration, which states that the integral of $x^n$ is $\frac{x^{n+1}}{n+1}$, where $n \neq -1$: $$ \int x^2 , dx = \frac{x^{2+1}}{2+1} = \frac{x^3}{3} $$
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Combine the Results Substituting back into the expression we factored earlier, we get: $$ \frac{1}{6} \cdot \frac{x^3}{3} = \frac{x^3}{18} $$
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Add the Constant of Integration Finally, don't forget to add the constant of integration $C$: $$ \frac{x^3}{18} + C $$
The final answer for the integral is: $$ \frac{x^3}{18} + C $$
More Information
This integral represents the area under the curve of the function $\frac{x^2}{6}$ with respect to $x$. The constant of integration $C$ is included since the antiderivative is not unique; it can take any constant value.
Tips
- Forgetting to add the constant of integration $C$ at the end of the integral.
- Incorrectly applying the power rule, especially with negative or fractional exponents.
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