To find the integral of x 2, written as \(\displaystyle \int x^2 \, dx\), we use the Power Rule for integration. The Power Rule for differentiation states:
\[ \dfrac{d}{dx}(x^n) = n \cdot x^{n-1} \]
To integrate, we reverse this process with what’s called the Reverse Power Rule:
\[ \displaystyle \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \]
For our example, we want to find \(\displaystyle \int x^2 \, dx\). Here, \(n = 2\), so we plug this into the formula:
\[ \displaystyle \int x^2 \, dx = \dfrac{x^{2+1}}{2+1} + C = \frac{x^3}{3} + C \]
So, the integral of \(x^2\) is $\dfrac{x^3}{3} + C$.
