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The question asks for the differential of an indefinite integral, specifically \( d\left(\int f(x) \, dx\right) \).

1. Let’s denote \( F(x) = \int f(x) \, dx \), which represents the antiderivative of \( f(x) \), meaning that \( F'(x) = f(x) \).

2. Taking the differential of \( F(x) \), we get \( dF(x) = F'(x) \, dx \). Since \( F'(x) = f(x) \), we have:
   \[
   d\left(\int f(x) \, dx\right) = f(x) \, dx.
   \]

Thus, the correct answer is **Option 2: \( f(x) \, dx \)**.

Would you like further details or have any questions?

Here are some related questions for deeper understanding:

1. What is the meaning of a differential in calculus?
2. How does the fundamental theorem of calculus relate to this problem?
3. What are the properties of indefinite integrals?
4. How would the answer change if we took the second differential?
5. Can differentials be applied to definite integrals in a similar way?

**Tip:** Remember that the differential of an integral essentially "undoes" the integral, resulting in the original integrand times \( dx \).

Чему равен дифференциал от неопределённого интеграла d(∫f(x)dx)?

Learn how to find the differential of an indefinite integral d(∫f(x)dx) with a step-by-step solution. This problem covers essential calculus concepts, such as differentiation and integration, and involves the Fundamental Theorem of Calculus. Suitable for high school students.

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