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To solve this integral problem, we need to analyze the given integral expression:

\[
\int_0^{\pi/4} \frac{\ln(\cot x)}{(\sin x)^{2009} + (\cos x)^{2009}} \cdot (\sin 2x)^{2008} \, dx = \frac{a^b \ln a}{c^2}
\]

where \( a \), \( b \), and \( c \) are integers in their lowest form. The goal is to find the value of \( a + b + c \).

### Step-by-Step Solution:

1. **Simplify \(\sin 2x\)**:
   - Recall that \( \sin 2x = 2 \sin x \cos x \).
   - Therefore, \( (\sin 2x)^{2008} = (2 \sin x \cos x)^{2008} = 2^{2008} (\sin x)^{2008} (\cos x)^{2008} \).

2. **Rewrite

If \( \int_0^{\pi/4} \frac{\ln(\cot x)}{(\sin x)^{2009} + (\cos x)^{2009}} \cdot (\sin 2x)^{2008} \, dx = \frac{a^b \ln a}{c^2} \), where \( a \), \( b \), and \( c \) are in their lowest form, find the value of \( a + b + c \).

This calculus problem requires evaluating an integral with logarithmic and trigonometric functions, using identities and integration techniques. Determine values of a, b, and c in lowest form, with a solution suitable for undergraduate students.

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