What is the integral of the function (x^2)/6 with respect to x?
Understand the Problem
The question is asking to compute the integral of the function (x^2)/6 with respect to x, which involves applying integration rules to determine the antiderivative of the given function.
Answer
$$ \frac{x^3}{18} + C $$
Answer for screen readers
The solution to the integral is $$ \frac{x^3}{18} + C $$.
Steps to Solve
- Identify the function to integrate
We are given the function $$ f(x) = \frac{x^2}{6} $$ that we need to integrate.
- Use the power rule of integration
The power rule states that the integral of $$ x^n $$ is given by $$ \frac{x^{n+1}}{n+1} + C $$ where $$ C $$ is the integration constant.
In our case, we can rewrite the function:
$$ \int f(x) , dx = \int \frac{x^2}{6} , dx = \frac{1}{6} \int x^2 , dx $$
Now we apply the power rule:
$$ \int x^2 , dx = \frac{x^{2+1}}{2+1} + C = \frac{x^3}{3} + C $$
- Combine results
Now putting it all together, we have:
$$ \int f(x) , dx = \frac{1}{6} \left(\frac{x^3}{3} + C\right) = \frac{x^3}{18} + C $$
Thus, the antiderivative of $$ \frac{x^2}{6} $$ is $$ \frac{x^3}{18} + C $$.
The solution to the integral is $$ \frac{x^3}{18} + C $$.
More Information
Integrating functions is a fundamental concept in calculus, used to find areas under curves and solve various problems related to accumulation and rates of change. The integration constant $$ C $$ represents the family of antiderivatives.
Tips
- Forgetting to include the integration constant $$ C $$ at the end of the integral.
- Not applying the power rule correctly, particularly in defining the new exponent.
