Limit of a Definite Integral 3

Calculus Level 5

\[\large \lim_{n\to\infty} \int_0^{n\pi} e^{-x} |\sin(x)| \, dx \]

If the limit above can be expressed as \( \alpha \left( \frac{1+e^\beta}{1-e^\lambda} \right) \) where \( \alpha, \beta, \lambda\) are real numbers, \(\alpha \geq 0\) and \( \beta, \lambda \leq 0\) and \(n\) is an integer, find the value of \(2(\alpha+\beta-\lambda)\).

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