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

  1. Identify the function to integrate

We need to integrate the function ( f(x) = \frac{x^2}{6} ).

  1. Set up the integral

We'll write the integral as: $$ \int \frac{x^2}{6} , dx $$

  1. Factor out the constant

The constant ( \frac{1}{6} ) can be factored out of the integral: $$ \frac{1}{6} \int x^2 , dx $$

  1. 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 ): $$ \int x^2 , dx = \frac{x^{2+1}}{2+1} + C = \frac{x^3}{3} + C $$

  1. Combine the results

Now, substitute back into the integral: $$ \frac{1}{6} \left( \frac{x^3}{3} + C \right) $$

  1. 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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