\( I =\int _{-\pi }^{\pi }\sin ^{2005} xdx \)
Đặt \(t=-x\).
Khi đó \(I =\int _{-\pi }^{\pi }\sin ^{2005} xdx =\int _{\pi }^{-\pi }\sin ^{2005} (-t)d(-t) \)
\(=\int _{\pi }^{-\pi }\sin ^{2005} tdt= -\int _{-\pi }^{\pi }\sin ^{2005} xdx =-I_{1} \)
Do đó \(I =\int _{-\pi }^{\pi }\sin ^{2005} xdx =0\)