Cosine and few friends

$\int_{-2016 \pi}^{2015 \pi} (\cos(x) + |\cos(x)|+ \cos(|x|)) dx\\=\int_{-2016 \pi}^{2015 \pi}\cos(x)dx+\int_{-2016 \pi}^{2015 \pi}|\cos(x)|dx+\int_{-2016 \pi}^{2015 \pi}\cos(|x|)dx\\=\left[\sin(x)\right]^{2015\pi}_{-2016\pi}+\int_{-2015.5 \pi}^{2015.5 \pi}|\cos(x)|dx+\int_{0}^{2015 \pi}\cos(x)dx+\int_{0}^{2016 \pi}\cos(x)dx\\=\left[\sin(x)\right]^{2015\pi}_{0}+\left[\sin(x)\right]^{2016\pi}_{0}+8062\int_{0}^{\frac{1}{2} \pi} \cos(x)dx\\=8062$

Let $I = \displaystyle \int_{-2016 \pi}^{2015 \pi} \cos x + |\cos x| + \cos |x| \ dx = I_1 + I_2 + I_3$ respectively.

We note that $cos x$ is an even function with a period or cycle of $2\pi$ and that $I_c = \displaystyle \int_a^b \cos x \ dx = 0$ , when $b-a=n\pi$ , where $n$ is an integer. Then we have:

$\begin{aligned} I_1 & = \int_{-2016 \pi}^{2015 \pi} \cos x \ dx = 0 & \small \color{#3D99F6} \text{Since }b-a = 4031\pi \end{aligned}$

$\begin{aligned} I_2 & = \int_{-2016 \pi}^{2015 \pi} |\cos x| \ dx & \small \color{#3D99F6} \text{All half-cycles are positive.} \\ & = \left(2015-(-2016)\right) \times 2\int_0^\frac \pi 2 \cos x \ dx \\ & = 8062 \sin x \bigg|_0^\frac \pi 2 \\ & = 8062 \end{aligned}$

$\begin{aligned} I_1 & = \int_{-2016 \pi}^{2015 \pi} \cos |x| \ dx = 0 & \small \color{#3D99F6} \text{Note that }\cos |x| = \cos x \text{ as both are even} \\ & = \int_{-2016 \pi}^{2015 \pi} \cos x \ dx = 0 & \small \color{#3D99F6} \text{Since }b-a = 4031\pi \end{aligned}$

Therefore, $I = I_1+I_2+I_3 = 0+8062+0 = \boxed{8062}$ .