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Đặt $t= \frac{\pi}{2}-x \Rightarrow dt=-dx, x=0 \rightarrow t= \frac{\pi}{2}, x= \frac{\pi}{2}\rightarrow t=0$
$I= \int_{0}^{\pi /2}\frac{\sin ^{2012}x}{\sin ^{2012}x + \cos ^{2012}x}dx$
$I= -\int_{\pi /2}^{0}\frac{\sin ^{2012}\left ( \frac{\pi}{2}-t \right )}{\sin ^{2012}\left ( \frac{\pi}{2}-t \right ) + \cos ^{2012}\left ( \frac{\pi}{2}-t \right )}dt$
$I= \int_{0}^{\pi /2}\frac{\cos ^{2012}t}{\cos ^{2012}t+ \sin ^{2012}t}dt$
$I= \int_{0}^{\pi /2}\left[ {1-\frac{\sin ^{2012}t}{\cos ^{2012}t+ \sin ^{2012}t}} \right]dt$
$I= \int_{0}^{\pi /2}dt-I$
$\boxed{I=\frac{\pi}{4}}$
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