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Đặt: $I=\int\limits_0^{\frac{\pi}{2}}\frac{\cos^{2010}x}{\sin^{2010}x+\cos^{2010}x}dx, J=\int\limits_0^{\frac{\pi}{2}}\frac{\sin^{2010}x}{\sin^{2010}x+\cos^{2010}x}dx $.
Ta có: $I+J=\int\limits_0^{\frac{\pi}{2}}dx=\frac{\pi}{2}$.
Đặt $t=\frac{\pi}{2}-x$. Suy ra:
$I=
\int\limits_{\frac{\pi}{2}}^0\frac{\cos^{2010}(
\displaystyle\frac{\pi}{2}-t)}{\sin^{2010}(\displaystyle\frac{\pi}{2}- t)+\cos^{2010}(\displaystyle \frac{\pi}{2}-t)}d(\frac{\pi}{2}-t)$
$=\int\limits_{\frac{\pi}{2}}^0\frac{-\sin^{2010}t}{\sin^{2010}t+\cos^{2010}t}dt$
$=\int\limits_0^{\frac{\pi}{2}}\frac{\sin^{2010}t}{\sin^{2010}t+\cos^{2010}t}dt=J$
Từ đó, suy ra: $I=J=\frac{\pi} {4}$ .
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