Here is the enhanced version of your content for improved clarity and readability:

The integral of ( x^2 , dx ) is given by

where ( C ) represents the constant of integration.

To evaluate the integral of ( x^2 , dx ), we apply the power rule of integration. This rule states that the integral of ( x^n , dx ) is

where ( C ) is also the constant of integration. By applying this rule to ( x^2 ), we find:

To verify our result, we can differentiate ( \frac{1}{3} x^3 + C ) with respect to ( x ). Using the power rule of differentiation, which states that the derivative of ( x^n ) is ( n x^{n-1} ), we differentiate ( \frac{1}{3} x^3 ):

This confirms that our integration result is correct.

In general, when evaluating integrals, it is crucial to remember the constant of integration because it can influence the final outcome. The constant arises from the fact that the derivative of any constant is zero. Therefore, adding a constant to the antiderivative does not change its derivative. Consequently, when performing integration, we always include the constant of integration to account for all possible antiderivatives.