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 𝑓𝑡𝑑𝑡=−1
 2+𝑥2+𝑥𝑠𝑒𝑛2𝑥 +1
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 2𝑥+𝑠𝑒𝑛2𝑥 +2𝑥𝑐𝑜𝑠2𝑥 −1
 22𝑠𝑒𝑛(2𝑥) 2𝑥+𝑠𝑒𝑛2𝑥 +2𝑥𝑐𝑜𝑠2𝑥 −𝑠𝑒𝑛(2𝑥)
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 Problema 4
 Hallar todos los valores de x
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 es correcto?

It looks like your text contains a mixture of mathematical expressions, trigonometric terms, and possibly some typos or formatting issues, so let me break it down step by step to see what's being asked and clarify.

One of the key parts of this appears to be asking for the solution to an integral equation involving \(x\), followed by some function evaluation and derivative calculations around \( \frac{1}{4\pi} \).

Let me clarify a few things based on what you provided:

### 1. Integral Evaluation:
You have written something like this:

\[
\int_0^x f(t) \, dt = -1 \quad \text{where} \quad f(x) = \frac{-1}{2 + x^2 + x \sin^2(x) + \frac{1}{2} \cos(2x)}
\]

I assume you are asking for an evaluation of \( f(x) \) and its derivative at \( x = \frac{1}{4\pi} \).

### 2. Function Evaluation \( f\left(\frac{1}{4\pi}\right) \):
It seems like you are trying to compute:

\[
f\left(\frac{1}{4\pi}\right) = \frac{-1}{2 + \left(\frac{1}{4\pi}\right)^2 + \left(\frac{1}{4\pi}\right) \sin^2\left(\frac{1}{4\pi}\right) + \frac{1}{2} \cos\left(2 \cdot \frac{1}{4\pi}\right)}
\]

This can be computed by substituting \( x = \frac{1}{4\pi} \) into the function and evaluating the terms.

### 3. Derivative \( f'\left(\frac{1}{4\pi}\right) \):
You also seem to be asking about the derivative of the function \( f(x) \) evaluated at \( \frac{1}{4\pi} \). The derivative involves applying the quotient rule and simplifying.

---

To clarify things more effectively, could you confirm if you're asking for:

1. The value of the integral for specific \( f(x) \)?
2. The evaluation of \( f\left(\frac{1}{4\pi}\right) \)?
3. The derivative of \( f(x) \) at \( x = \frac{1}{4\pi} \)?
4. Solutions for \( x \) in the problem where the integral equation is given?

Let me know which part you'd like me to prioritize or if you'd like a full breakdown of these steps.

### Expanding Knowledge:

1. What is the integral form of \( f(t) \) you're working with?
2. Are there any specific initial conditions for the integral equation?
3. What are the constraints or domains of \( x \)?
4. How are the trigonometric terms expected to behave (e.g., sine and cosine)?
5. Is the problem asking for a general solution for all \( x \)?

### Tip:
For derivative problems with trigonometric and polynomial functions, always apply the chain rule and quotient rule carefully.

න 0 𝑥 𝑓𝑡𝑑𝑡=−1 2+𝑥2+𝑥𝑠𝑒𝑛2𝑥 +1 2 cos(2𝑥) 𝑓(1 4𝜋) 𝑓′(1 4𝜋) 𝑓(1 4𝜋) −1 2+ 1 4𝜋 2 + 1 4𝜋 𝑠𝑒𝑛 21 4𝜋 +1 2 cos(21 4𝜋) −1 2+1 16𝜋2+1 4𝜋𝑠𝑒𝑛 𝜋 2 +1 2 cos(𝜋 2) −1 2+1 16𝜋2+1 4𝜋1+1 2 (0) −1 2+1 16𝜋2+1 4𝜋 𝑓′(1 4𝜋) 2𝑥+𝑠𝑒𝑛2𝑥 +2𝑥𝑐𝑜𝑠2𝑥 −1 22𝑠𝑒𝑛(2𝑥) 2𝑥+𝑠𝑒𝑛2𝑥 +2𝑥𝑐𝑜𝑠2𝑥 −𝑠𝑒𝑛(2𝑥) 1 1 1 1 1 1 2 1 1 4𝜋 +𝑠𝑒𝑛 2𝜋+𝑠𝑒𝑛 2 4𝜋 +2 𝜋 2 +𝜋 2 cos Problema 4 Hallar todos los valores de x 𝑥

Explore the detailed steps to solve the integral and derivative of the trigonometric function f(x) involving sine and cosine terms. This problem covers advanced calculus and trigonometry, including the application of the quotient rule and the evaluation at specific points like 1/4π.

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