The image you've uploaded shows the indefinite integral:

$\int dx$

This integral represents the most basic form of integration in calculus. Let's break it down step by step:

Understanding the Integral

1. Indefinite Integral:

   • The symbol $\int$ represents the operation of integration.
   • The term $dx$ indicates that we are integrating with respect to the variable $x$.
   • Since there is no specific function shown inside the integral (i.e., there is no function being integrated), it implies the integration of the constant $1$ with respect to $x$.

2. The Integral of 1:

   • Mathematically, the integral can be expressed as: $\int 1 \, dx$
   • The task is to find a function $F(x)$ whose derivative $F'(x)$ is equal to $1$.

3. Finding the Antiderivative:

   • The function whose derivative is $1$ is $x$ because the derivative of $x$ with respect to $x$ is: $\frac{d}{dx}(x) = 1$
   • Therefore, the integral of $1$ with respect to $x$ is $x$.

4. Adding the Constant of Integration:

   • An indefinite integral includes a constant of integration, denoted by $C$.
   • This constant $C$ is added because integration is the inverse operation of differentiation, and differentiating any constant results in zero.
   • Thus, the general solution to the integral $\int 1 \, dx$ is: $x + C$

Complete Solution

Thus, the result of the integral is:

$\int dx = x + C$

Why the Constant $C$?

• When you integrate a function, you are essentially reversing the process of differentiation. Since the derivative of a constant is zero, the original function could have been $x$, $x + 1$, $x + 2$, etc.
• For instance, if $F(x) = x + 3$, then $F'(x) = 1$, just like it is for $F(x) = x$.
• The constant $C$ represents all possible vertical shifts of the function that yield the same derivative.

Summary

• The integral $\int dx$ means finding a function whose derivative is $1$.
• The answer is $x + C$ where $C$ is an arbitrary constant, accounting for all possible functions that have a derivative of $1$.

Do you have any questions or want more details?

Here are some related questions to expand on this topic:

1. What is the geometric interpretation of the integral $\int dx$?
2. How do definite integrals differ from indefinite integrals?
3. Why is the constant of integration important in indefinite integrals?
4. How does the power rule for integration apply to polynomials?
5. What are some applications of integration in real-world problems?
6. How would you evaluate an integral with specific limits of integration?
7. How can you check your integration result for correctness?
8. What are common mistakes to avoid when integrating functions?

Tip: When integrating, always remember to add the constant of integration $C$ for indefinite integrals to account for all possible functions that could result from the integration.