The value of the integral $\int\limits_a^{a+\pi / 2}(|\sin x|+|\cos x|) d x$, is

independent of a

Since $f(x)=|\sin x|+|\cos x|$ is a periodic function with period $\frac{\pi}{2}$. Therefore, $\int\limits_a^{a+\pi / 2} f(x) d x$ is independent of $a$. In fact, we have

$\int\limits_a^{a+\pi / 2} f(x) d x=\int\limits_0^{\pi / 2} f(x) d x=\int\limits_0^{\pi / 2}(|\sin x|+|\cos x|) d x$

$\Rightarrow \int\limits_a^{a+\pi / 2} f(x) d x=\int\limits_0^{\pi / 2}(\sin x+\cos x) d x$