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The given problem asks for the length of the curve defined by:

\[
y = \int_0^x \cos(2t) \, dt
\]
from \( x = 0 \) to \( x = \frac{\pi}{4} \).

### Step 1: Find \( \frac{dy}{dx} \)

Using the Fundamental Theorem of Calculus, the derivative of \( y = \int_0^x \cos(2t) \, dt \) with respect to \( x \) is:

\[
\frac{dy}{dx} = \cos(2x)
\]

### Step 2: Set up the formula for the length of the curve

The formula for the arc length \( L \) of a curve \( y = f(x) \) from \( x = a \) to \( x = b \) is:

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

In this case, \( \frac{dy}{dx} = \cos(2x) \), and we are evaluating from \( x = 0 \) to \( x = \frac{\pi}{4} \). Thus, the integral becomes:

\[
L = \int_0^{\frac{\pi}{4}} \sqrt{1 + \cos^2(2x)} \, dx
\]

This is the exact form of the integral that represents the length of the curve.

Would you like to proceed with solving this integral, or is this setup sufficient?

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Here are 5 related questions to further explore this topic:
1. How does the Fundamental Theorem of Calculus apply to finding derivatives of integral functions?
2. Can you calculate the derivative of an integral where the limits of integration are not constants?
3. How does the arc length formula change when the curve is given parametrically?
4. What are some common techniques for solving integrals involving trigonometric functions like \( \cos^2(x) \)?
5. How do you use numerical methods to approximate the length of a curve if the integral cannot be solved analytically?

### Tip:
When calculating arc lengths, simplifying the integrand first can often make solving the integral easier!

Find the length of the curve y = ∫₀ˣ cos(2t) dt from x = 0 to x = π/4.

Learn how to calculate the length of a curve defined by y = ∫₀ˣ cos(2t) dt, applying the arc length formula and the Fundamental Theorem of Calculus. This step-by-step solution is suitable for college-level calculus students.

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