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 applying integration techniques to find the antiderivative of this polynomial expression.
Answer
The final answer 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
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Identify the Integral
We need to find the integral of the function $\frac{x^2}{6}$. This can be written as:
$$ \int \frac{x^2}{6} , dx $$ -
Extract Constants from the Integral
We can factor out the constant $\frac{1}{6}$ from the integral:
$$ \frac{1}{6} \int x^2 , dx $$ -
Apply the Power Rule of Integration
According to the power rule, the integral of $x^n$ is $\frac{x^{n+1}}{n+1} + C$. Here, $n = 2$:
$$ \int x^2 , dx = \frac{x^{2+1}}{2+1} + C = \frac{x^3}{3} + C $$ -
Combine the Results
Now, substitute the result back into the equation:
$$ \frac{1}{6} \cdot \left( \frac{x^3}{3} + C \right) $$ -
Final Simplification
Multiply through by $\frac{1}{6}$:
$$ \frac{x^3}{18} + \frac{C}{6} $$
We can denote $\frac{C}{6}$ simply as $C$ because it's still an arbitrary constant. Thus, we have:
$$ \frac{x^3}{18} + C $$
The integral of the function $\frac{x^2}{6}$ with respect to $x$ is:
$$ \frac{x^3}{18} + C $$
More Information
This result is part of the fundamental concept of calculus known as integration, which allows us to find the antiderivative or area under the curve of a function. Integration is widely used in various fields such as physics, engineering, and economics.
Tips
- Forgetting to include the constant of integration $C$ at the end. Always remember to add this unless specified otherwise.
- Misapplying the power rule, such as using the wrong exponent when integrating.
