Find the area between the graph $y=e^{-x}\sin x, x \geq 0$ and the $x$-axis. Calculate the area of the area.
answer：
The part of the red line is marked out, how do you think of it? If you don’t read the answer, what do you do? I can’t think of this answer. I don’t understand it. It may be that I lack some knowledge and ask God to give pointers on background and facts.
They have calculated $$J_k:=\int_{k\pi}^{(k+1)\pi}e^{-x}|\sin x|\>dx$$ for even and odd $k$ separately, and have obtained $$J_k={1\over2}e^{-k\pi}(1+e^{-\pi})$$ in both cases. It follows that $$S=\int_0^\infty e^{-x}|\sin x|\>dx=\sum_{k=0}^\infty J_k={1\over2}(1+e^{-\pi})\sum_{k=0}^\infty e^{-k\pi}={1\over2}\,{1+e^{-\pi}\over 1-e^{-\pi}}\ .$$