can you show me what differentiation rule u use to solve this 

To solve the integral \(\int dx\), we use the fact that differentiation and integration are inverse operations. Here, we need to find a function \(F(x)\) such that \(F'(x) = 1\). 

The differentiation rule we use is the power rule for derivatives. For a function \(x^n\), the derivative is given by:

\[
\frac{d}{dx} x^n = n x^{n-1}
\]

In this case, we consider \(n = 1\):

\[
\frac{d}{dx} x = 1
\]

Thus, the function \(x\) has a derivative of 1. Therefore, the integral of 1 with respect to \(x\) is \(x\), plus a constant of integration \(C\). This gives us:

\[
\int dx = x + C
\]

Would you like any more details or have any questions about this solution?

Here are 8 related questions for further exploration:

1. How do you apply the power rule for integration?
2. What is the integral of \(x^3\) with respect to \(x\)?
3. Can you explain the product rule for differentiation?
4. How do you find the integral of \(a\) with respect to \(x\), where \(a\) is a constant?
5. What is the chain rule for differentiation?
6. How do you solve the integral \(\int (2x+3) dx\)?
7. What is the integral of \(\frac{1}{x}\) with respect to \(x\)?
8. How do you differentiate \(e^{x^2}\)?

**Tip:** To verify your integration results, you can differentiate the antiderivative. If you get the original function back, your integration is correct.

Explore how differentiation and integration are inverse operations, focusing on the power rule. Learn the key formula for derivatives \( \frac{d}{dx} x^n = n x^{n-1} \) and apply it to solve integration problems. Suitable for students in Grades 11-12.

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