Certainly! Let's delve into the problem step-by-step.

When you see the integral $\int dx$, it's asking for the antiderivative of $1$ with respect to $x$. Here's the detailed reasoning:

1. Understanding the Problem: $\int dx$ This integral represents the antiderivative of the function $f(x) = 1$.

2. Antiderivative Concept: The antiderivative of a function $f(x)$ is a function $F(x)$ such that $F'(x) = f(x)$.

3. Applying the Concept: For $f(x) = 1$, we need a function $F(x)$ whose derivative is $1$.

4. Finding the Function $F(x)$:

   • Consider the function $F(x) = x$.
   • The derivative of $x$ with respect to $x$ is: $\frac{d}{dx}(x) = 1$ Therefore, $F(x) = x$ is an antiderivative of $1$.

5. Adding the Constant of Integration:

   • The integral $\int dx$ actually represents a family of functions.
   • Since differentiation of a constant is zero, we include an arbitrary constant $C$ in our answer to represent all possible antiderivatives.
   • So, the most general form of the antiderivative is: $\int dx = x + C$

6. Summary:

   • $\int dx = x + C$
   • This means that the antiderivative of $1$ with respect to $x$ is $x$, plus an arbitrary constant $C$.

Steps Recap:

1. Recognize the integral: $\int dx$.
2. Identify the function being integrated: $f(x) = 1$.
3. Determine a function whose derivative is $1$: $F(x) = x$.
4. Include the constant of integration $C$.

Tip:

When you integrate a constant function $c$, the result is $cx + C$, where $C$ is the constant of integration.

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Do you have any questions or would you like further details on any part of this process?

Related Questions:

1. What is the antiderivative of $2$?
2. How do you integrate a function $f(x) = k$ where $k$ is a constant?
3. Can you explain the difference between definite and indefinite integrals?
4. How do you find the derivative of a polynomial function?
5. What is the Fundamental Theorem of Calculus?
6. How do you integrate $f(x) = x$?
7. What is the constant of integration and why is it important?
8. How do you check if a function is the correct antiderivative?

Tip:

Always remember to add the constant of integration $C$ when solving indefinite integrals, as it represents an entire family of possible solutions.