To solve this integral using the Power Rule, let's first express the integral clearly:

$\int u^2 \, du$

The Power Rule for integration states that for any constant $n \neq -1$,

$\int u^n \, du = \frac{u^{n+1}}{n+1} + C$

Now, applying this rule to $u^2$:

$\int u^2 \, du = \frac{u^{2+1}}{2+1} + C = \frac{u^3}{3} + C$

So, the result of the integral is:

$\frac{u^3}{3} + C$

Now, you also mentioned an expression involving $u^{-2}$, so let's consider the following:

$\int u^{-2} \, du$

Using the Power Rule for this case:

$\int u^{-2} \, du = \frac{u^{-2+1}}{-2+1} + C = \frac{u^{-1}}{-1} + C = -\frac{1}{u} + C$

So, the result for $\int u^{-2} \, du$ is:

$-\frac{1}{u} + C$

Let me know if you'd like more details or further clarification!

Follow-up Questions:

1. How does the Power Rule for integration apply to rational functions like $\int \frac{1}{u^2} \, du$?
2. Can you differentiate $\frac{u^3}{3} + C$ and explain why it matches the original function $u^2$?
3. What happens when $n = -1$ in the Power Rule? Why does the formula fail in that case?
4. Can you integrate $\int u^n \, du$ when $n = -1$ using a different method?
5. How does the constant of integration $C$ influence the solution when applying the Power Rule?

Tip: Remember to always check the exponent of $u$ to ensure the Power Rule is applied correctly. For $n = -1$, you’ll need to handle it as a special case!