Application Of Integration

Consider the curve $C_1$ given by

\(y=e^{-x} \quad \text { for } x \in[0,10 \pi],\)

and the curve $C_2$ given by

\(y=e^{-x}(\sin x+\cos x) \quad \text { for } x \in[0,10 \pi] .\)

Let $n$ be the total number of points of intersection of the curves $C_1$ and $C_2$.

Suppose that $\alpha_1, \alpha_2, \ldots, \alpha_n \in[0,10 \pi]$ are the $x$-coordinates of the points of intersection of the curves $C_1$ and $C_2$ such that

\(\alpha_1<\alpha_2<\cdots<\alpha_n .\)

Let $\beta$ be the area of the region enclosed between the curves $C_1, C_2$, and the lines $x=\alpha_1$ and $x=\alpha_4$. Then the value of

\(-\frac{1}{\pi} \log _e\left(\beta-2 e^{-\frac{\pi}{2}}\right)\)

is $\_\_\_\_$ .