What is the integral of (x^2)/6 with respect to x?
Understand the Problem
The question is asking us to find the integral of the function (x^2)/6 with respect to x, which involves applying integration rules to determine the antiderivative of this function.
Answer
The integral of the function $\frac{x^2}{6}$ with respect to $x$ is $\frac{x^{3}}{18} + C$.
Answer for screen readers
The integral of the function $\frac{x^2}{6}$ with respect to $x$ is
$$ \frac{x^{3}}{18} + C $$
Steps to Solve
- Identify the Function to Integrate
We are given the function to integrate: $$ f(x) = \frac{x^2}{6} $$
- Apply the Power Rule of Integration
Using the power rule for integration, which states that $$ \int x^n , dx = \frac{x^{n+1}}{n+1} + C $$ for any real number n ≠ -1.
In our case, n is 2, thus applying the rule:
$$ \int \frac{x^2}{6} , dx = \frac{1}{6} \cdot \int x^2 , dx $$
- Integrate the Function
Now integrating $x^2$ using the power rule:
$$ \int x^2 , dx = \frac{x^{3}}{3} + C $$
So,
$$ \int \frac{x^2}{6} , dx = \frac{1}{6} \cdot \left( \frac{x^{3}}{3} + C \right) $$
- Simplify the Result
Now we simplify the expression:
$$ \int \frac{x^2}{6} , dx = \frac{x^{3}}{18} + C $$
The integral of the function $\frac{x^2}{6}$ with respect to $x$ is
$$ \frac{x^{3}}{18} + C $$
More Information
This integral helps in various fields such as physics, particularly in finding areas under curves or when calculating motion. Understanding integration and its rules is fundamental for handling more complex calculus problems.
Tips
- Misapplying the power rule, such as forgetting to add 1 to the exponent.
- Forgetting to include the constant of integration (C) after performing the integral.
