Subject: Mathematics — Definite Integration
Question: Consider the curve $C_{1}$ given by
$$y = e^{- x}\text{~for~}x \in \lbrack 0,10\pi\rbrack$$
and the curve $C_{2}$ given by
$$y = e^{- x}(sinx + cosx)\text{~for~}x \in \lbrack 0,10\pi\rbrack$$
Let $n$ be the total number of points of intersection of the curves
$C_{1}$ and $C_{2}$.\
Suppose that
$\alpha_{1},\alpha_{2},\ldots,\alpha_{n}, \in \lbrack 0,10\pi\rbrack$
are the $x$ - coordinates of the points of intersection of the curves
$C_{1}$ and $C_{2}$ such that
$$\alpha_{1} < \alpha_{2} < \ldots < \alpha_{n}.$$
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\
$- \frac{1}{\pi}\log_{e}\left( \beta - 2e^{- \frac{\pi}{2}} \right)$
is $\_\_\_\_$ .
Answer: 2.50
Solution: $\alpha_{1} = 0\&\alpha_{4} = \frac{5\pi}{2}$
[IMAGE] $$\beta = \int_{0}^{\frac{5\pi}{2}}\mspace{2mu}\mspace{2mu}\left| \left( e^{- x}(sinx + cosx) - e^{- x} \right) \right|dx$$
$$\beta = \int_{0}^{\frac{\pi}{2}}\mspace{2mu}\left( e^{- x}(sinx + cosx) - e^{- x} \right)dx + \int_{\frac{\pi}{2}}^{2\pi}\mspace{2mu}\left( e^{- x} - e^{- x}(sinx + cosx) \right)dx$$
$$+ \int_{2\pi}^{\frac{5\pi}{2}}\mspace{2mu}\mspace{2mu}\left( e^{- x}(sinx + cosx) - e^{- x} \right)dx$$
$$\beta = 2e^{\frac{- \pi}{2}} + e^{\frac{- 5\pi}{2}}$$
$$\text{so~}\frac{- 1}{\pi}log\left( \beta - 2e^{\frac{- \pi}{2}} \right) = \frac{5}{2} = 2.50$$