Evaluate: pie/2 integrate -pie/2 |sin x + cos x| dx\(\int\limits_{\pi/

1.	Evaluate: pie/2 integrate -pie/2 \|sin x + cos x\| dx\(\int\limits_{\pi/2}^{\pi/2}\)\|sin x + cos x\| dx.
Answer» 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\(\sqrt2\)

Evaluate: pie/2 integrate -pie/2 |sin x + cos x| dx\(\int\limits_{\pi/2}^{\pi/2}\)|sin x + cos x| dx.

Answer»

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\(\sqrt2\)