Exponential Integral Involving Absolute Sine Function

The integral \({{I}_{n}}\) is defined by

\[ I_n = \displaystyle\displaystyle \displaystyle\int_{n\pi}^{(n+1)\pi} e^{-x}\,|\sin x|\,dx, for \quad n \in \mathbb{N}. \]

Show that

\[ I_0 = \frac{1}{2}\left( 1 + e^{-\pi} \right). \][6]

By letting \(y = x - n\pi\), show that

\[ I_n = e^{-n\pi} I_0. \][4]

Hence determine the exact value of \(\displaystyle\displaystyle \displaystyle\int_{0}^{\infty }{{{\mathrm{e}}^{-x}}\left| \sin x \right|}\mathrm{ d}x\).[5]