Sine and few friends

Let $I = \displaystyle \int_{-2017 \pi}^{2018 \pi} \sin x + |\sin x| + \sin |x| \ dx = I_1 + I_2 + I_3$ respectively.

We note that $sin x$ has a period or cycle of $2\pi$ . As an odd function, its area under the curve over one cycle is 0 and half a cycle has positive and the other half has equal area but negative. The absolute value of area of half a cycle is $\displaystyle \int_0^\pi \sin x \ dx = - \cos x \bigg|_0^\pi = 2$ . Then we have $\displaystyle I_s(n) = \int_0^{n\pi} \sin x \ dx = \begin{cases} 2 & \text{if }n \text{ is odd} \\ 0 & \text{if }n \text{ is even} \end{cases} \quad \forall \ n \in \mathbb N$

$\begin{aligned} I_1 & = \int_{-2017\pi}^{2018 \pi} \sin x \ dx \\ & = {\color{#3D99F6}\int_{-2017\pi}^0 \sin x \ dx} + \int_0^{2018 \pi} \sin x \ dx & \small \color{#3D99F6} \text{Since }\sin x \text{ is odd} \\ & = {\color{#3D99F6}- \int_0^{2017\pi} \sin x \ dx} + I_s(2018) \\ & = - I_s(2017) + 0 \\ & = - 2 \end{aligned}$

$\begin{aligned} I_2 & = \int_{-2017\pi}^{2018 \pi} |\sin x| \ dx & \small \color{#3D99F6} \text{All half-cycles are positive.} \\ & = \left(2018-(-2017)\right) I_s(1) \\ & = 8070 \end{aligned}$

$\begin{aligned} I_3 & = \int_{-2017\pi}^{2018 \pi} \sin |x| \ dx \\ & = {\color{#3D99F6}\int_{-2017\pi}^0 \sin |x| \ dx} + \int_0^{2018 \pi} \sin |x| \ dx & \small \color{#3D99F6} \text{Since }\sin |x| \text{ is even} \\ & = {\color{#3D99F6}\int_0^{2017\pi} \sin x \ dx} + \int_0^{2018 \pi} \sin x \ dx \\ & = I_s(2017) + I_s(2018) \\ & = 2 \end{aligned}$

Therefore, $I = I_1+I_2+I_3 = -2+8070+2 = \boxed{8070}$ .