This is question that was tested in ACJC H2 Math Prelim P1 2015. A few of my students know how to answer it but were uncertain how to express it.

Most students were concerned with (ii) of the question.

I think the easiest way to prove this, is to first avoid writing too much and attempt to show it mathematically.

$\int_{\pi /3}^{2\pi /3} |cos\frac{x}{2} cosx| dx$
$= \int_{\pi /3}^{pi /2} cos\frac{x}{2} cosx dx - \int_{\pi /2}^{2\pi /3} cos\frac{x}{2} cosx dx$

while

$|\int_{\pi /3}^{2\pi /3} cos\frac{x}{2} cosx dx|$
$= |\int_{\pi /3}^{pi /2} cos\frac{x}{2} cosx dx - \int_{\pi /2}^{2\pi /3} cos\frac{x}{2} cosx dx|$

Thus $|\int_{\pi /3}^{2\pi /3} cos\frac{x}{2} cosx dx|$ will be smaller in magnitude.

Students can also attempt to justify using the area under graph but they must express the answers in words carefully.

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