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 requires applying the rules of integration to find the antiderivative of the given function.
Answer
The integral of \( \frac{x^2}{6} \) is \( \frac{x^3}{18} + C \).
Answer for screen readers
The integral of ( \frac{x^2}{6} ) with respect to ( x ) is given by: $$ \frac{x^3}{18} + C $$
Steps to Solve
- Identify the function to integrate
We need to integrate the function ( f(x) = \frac{x^2}{6} ).
- Set up the integral
We'll write the integral as: $$ \int \frac{x^2}{6} , dx $$
- Factor out the constant
The constant ( \frac{1}{6} ) can be factored out of the integral: $$ \frac{1}{6} \int x^2 , dx $$
- Integrate the power of x
Using the power rule of integration, which states that ( \int x^n , dx = \frac{x^{n+1}}{n+1} + C ), we apply it to ( x^2 ): $$ \int x^2 , dx = \frac{x^{2+1}}{2+1} + C = \frac{x^3}{3} + C $$
- Combine the results
Now, substitute back into the integral: $$ \frac{1}{6} \left( \frac{x^3}{3} + C \right) $$
- Simplify the expression
Distributing ( \frac{1}{6} ): $$ \frac{x^3}{18} + \frac{C}{6} $$
Since ( \frac{C}{6} ) is still a constant, we can express it as a constant ( C' ): $$ \frac{x^3}{18} + C' $$
The integral of ( \frac{x^2}{6} ) with respect to ( x ) is given by: $$ \frac{x^3}{18} + C $$
More Information
The process of finding the integral, or antiderivative, of a function is crucial in calculus. Integration is often used in physics and engineering to find areas, volumes, or accumulated quantities.
Tips
- Forgetting to include the constant of integration ( C ) at the end of the integral.
- Not properly applying the power rule for integration. Be sure to increment the exponent and divide by the new exponent.
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