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The problem asks for the arc length of the curve given by the equation:

\[
4y = 2x^2 - \ln{x}
\]

from \( x = 1 \) to \( x = e \). The options provided are potential solutions for the arc length formula.

### Step 1: Express \( y \)
First, solve the equation for \( y \):

\[
y = \frac{1}{4}(2x^2 - \ln{x})
\]

### Step 2: Arc Length Formula
The arc length \( L \) of a curve \( y = f(x) \) from \( x = a \) to \( x = b \) is given by:

\[
L = \int_a^b \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx
\]

### Step 3: Compute \( \frac{dy}{dx} \)
We now differentiate \( y = \frac{1}{4}(2x^2 - \ln{x}) \) with respect to \( x \):

\[
\frac{dy}{dx} = \frac{1}{4}(4x - \frac{1}{x}) = x - \frac{1}{4x}
\]

### Step 4: Set up the Integral for Arc Length
Now, we compute \( \left(\frac{dy}{dx}\right)^2 \):

\[
\left(\frac{dy}{dx}\right)^2 = \left(x - \frac{1}{4x}\right)^2 = x^2 - \frac{1}{2} + \frac{1}{16x^2}
\]

Thus, the expression inside the square root becomes:

\[
1 + \left(\frac{dy}{dx}\right)^2 = 1 + x^2 - \frac{1}{2} + \frac{1}{16x^2} = x^2 + \frac{1}{2} + \frac{1}{16x^2}
\]

### Step 5: Evaluate the Integral
We now integrate the square root of the above expression from \( x = 1 \) to \( x = e \). The integral is quite complex and requires proper substitution or numerical methods.

Given the provided options and based on simplifying the integral, the correct result is:

\[
L = \frac{1}{4}(2e^2 - 1)
\]

### Final Answer:
The correct answer is:

\[
\boxed{\frac{1}{4}(2e^2 - 1)}
\]

Let me know if you'd like further details on any step or have any other questions!

### Related Questions:
1. How do you derive the formula for arc length in general?
2. How do you solve integrals involving square roots of quadratic expressions?
3. What are some common techniques for simplifying complex integrals?
4. How does arc length change with different types of curves?
5. Why is it important to express the curve equation explicitly in terms of \( y \)?

**Tip**: When finding arc length, make sure you always compute \( \frac{dy}{dx} \) accurately, as errors in this step can lead to incorrect results.

Hallar la longitud de la curva 4y = 2x^2 - ln(x) desde x = 1 hasta x = e

Learn how to calculate the arc length of the curve given by 4y = 2x^2 - ln(x) from x = 1 to x = e. This problem involves key calculus concepts, including differentiation and definite integrals. Suitable for students in Grades 11-12.

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