I =\(\int\limits_{\pi/2}^{\pi/2}\)|sin x + cos x|dx
= \(\int\limits_{-\pi/2}^{-\pi/4}\)-(sin x + cos x)dx + \(\int\limits_{-\pi/4}^{\pi/2}\)(sin x + cos x)dx
= \(-[-cos x + sin x]_{-\pi/2}^{-\pi/4}dx\) + \([-cos x+sin x]_{-\pi/4}^{\pi/2}\)
= -(-cos π/4 - sin π/4) + (-cos π/2 - sin π/2) + (- cos π/2 + sin π/2) - (-cos π/4 - sin π/4)
(\(\because\) cos(-θ) + cos θ and sin(-θ) = -sin θ)
= cos π/4 + sin π/4 - cos π/2 - sin π/2 - cos π/2 + sin π/2 + cos π/4 + sin π/4
= 2 cos π/4 + 2 sin π/4 - 2 cos π/2
= 2 x \(\frac1{\sqrt2}\) + 2 x \(\frac1{\sqrt2}\) - 2 x 0
= \(\sqrt2+\sqrt2\)
= 2\(\sqrt2\)
