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 techniques to obtain the antiderivative of the function.
Answer
The integral 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 integral to be solved
You need to find the integral of the function (\frac{x^2}{6}). This can be written as: $$ \int \frac{x^2}{6} , dx $$
- Factor out the constant
Since (\frac{1}{6}) is a constant, you can factor it out of the integral. This gives: $$ \frac{1}{6} \int x^2 , dx $$
- Apply the power rule of integration
The power rule for integration states that: $$ \int x^n , dx = \frac{x^{n+1}}{n+1} + C $$ where (C) is the constant of integration. In this case, (n = 2): $$ \int x^2 , dx = \frac{x^{2+1}}{2+1} + C = \frac{x^3}{3} + C $$
- Substitute back to find the final integral
Now substitute back into the equation: $$ \frac{1}{6} \left( \frac{x^3}{3} + C \right) = \frac{x^3}{18} + \frac{C}{6} $$
- Write final form including constant of integration
The final result of the integral can be written as: $$ \frac{x^3}{18} + C $$
The integral of the function (\frac{x^2}{6}) with respect to (x) is: $$ \frac{x^3}{18} + C $$
More Information
The result represents the area under the curve of the function (\frac{x^2}{6}) with respect to the x-axis. Integrals are fundamental in calculus, as they are used to calculate areas, volumes, and in solving differential equations.
Tips
- Forgetting to include the constant of integration (C) after performing the integration.
- Not factoring out constants properly before applying the power rule.
