To solve the integral equation $u(x) = x + \lambda \int_{0}^{2\pi} |\sin(u(t))| \, dt$ using the method of degenerate kernels, let's proceed step by step:

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Step 1: Represent the kernel as a degenerate kernel

In the degenerate kernel method, we approximate the kernel or simplify the integral so that it can be expressed in terms of a finite sum. The kernel here is not explicitly separable, but let us assume that $|\sin(u(t))|$ can be expressed in terms of an expansion or approximated for simplicity.

Step 2: Consider properties of $|\sin(u(t))|$

The function $|\sin(u(t))|$ is periodic with period $2\pi$, and its integral over $[0, 2\pi]$ can be computed independently:

[ \int_{0}^{2\pi} |\sin(u(t))| , dt = 2 \int_{0}^{\pi} \sin(t) , dt = 2 \cdot [-\cos(t)]_{0}^{\pi} = 2 \cdot [1 - (-1)] = 4. ]

Thus, the kernel simplifies to a constant term.

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Step 3: Substitute into the integral equation

The equation becomes:

$u(x) = x + \lambda \cdot 4.$

This indicates that $u(x)$ is independent of $t$ due to the constancy of the integral. Rewrite the solution as:

$u(x) = x + 4\lambda.$

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Step 4: Verify the solution

Plug $u(x) = x + 4\lambda$ back into the original equation to confirm consistency:

1. Compute $u(t) = t + 4\lambda$, and substitute this into $|\sin(u(t))|$.
2. Note that $|\sin(t + 4\lambda)|$ still integrates over $[0, 2\pi]$ to a constant factor due to periodicity, confirming the simplification.

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Final Solution

The solution to the integral equation is:

$u(x) = x + 4\lambda.$

Would you like further details or steps about this method? Here are some follow-up questions:

1. Can we explore the periodicity of $|\sin(u(t))|$ in more detail?
2. Would you like an explanation about the degenerate kernel method for non-constant cases?
3. How does the result change if $|\sin(u(t))|$ is replaced with another kernel?
4. Would you like to compute specific examples for $\lambda$?
5. Do you want to learn how to solve similar integral equations?

Tip: Periodic functions often simplify integral equations due to their predictable patterns over specific intervals. Always check for periodicity!