What is the integral of (x^2)/6 with respect to x?
Understand the Problem
The question is asking to find the integral of the function (x^2)/6 with respect to x, which involves applying the rules of integration to obtain the antiderivative.
Answer
$$ \int \frac{x^2}{6} \, dx = \frac{x^{3}}{18} + C $$
Answer for screen readers
$$ \int \frac{x^2}{6} , dx = \frac{x^{3}}{18} + C $$
Steps to Solve
-
Set up the integral We want to find the integral of the function ( \frac{x^2}{6} ) with respect to ( x ). This can be written as: $$ \int \frac{x^2}{6} , dx $$
-
Factor out the constant Since ( \frac{1}{6} ) is a constant, we can factor it out of the integral to simplify the calculation: $$ = \frac{1}{6} \int x^2 , dx $$
-
Apply the power rule of integration The power rule states that: $$ \int x^n , dx = \frac{x^{n+1}}{n+1} + C $$ For our function, ( n = 2 ): $$ = \frac{1}{6} \left( \frac{x^{2+1}}{2+1} \right) + C $$
-
Simplify the expression Now, we can write the integral with the calculated exponents and simplify: $$ = \frac{1}{6} \left( \frac{x^{3}}{3} \right) + C $$ This becomes: $$ = \frac{x^{3}}{18} + C $$
-
Write the final answer Finally, include the constant of integration ( C ): $$ \int \frac{x^2}{6} , dx = \frac{x^{3}}{18} + C $$
$$ \int \frac{x^2}{6} , dx = \frac{x^{3}}{18} + C $$
More Information
The integral of a polynomial function like ( \frac{x^2}{6} ) can be found using basic integration rules such as the power rule. This result allows us to find the area under the curve of the function with respect to the variable ( x ).
Tips
- Forgetting to add the constant of integration ( C ) at the end of the integration process. Always remember that antiderivatives include an arbitrary constant.
