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Question

$I=\int\limits_{0}^{\pi}\frac{\sin^{2014}x}{\sin^{2014}x+\cos^{2014}x}dx$

score 2 · Answer 1

$I = \int\limits_{0}^{\pi}\frac{\sin^{2014}x}{\sin^{2014}x+\cos^{2014}x}dx =\int\limits_{0}^{\pi/2}\frac{\sin^{2014}x}{\sin^{2014}x+\cos^{2014}x}dx +\int\limits_{\pi/2}^{\pi}\frac{\sin^{2014}x}{\sin^{2014}x+\cos^{2014}x}dx = I_1+I_2$

Xét

$I_2$

Đặt

$x = \pi/2+t$

$dx = dt$

$x = \pi/2 \to t = 0; x = \pi \to t = \pi/2$

$I_2 = \int\limits_{\pi/2}^{\pi}\frac{\sin^{2014}x}{\sin^{2014}x+\cos^{2014}x}dx=\int\limits_{0}^{\pi/2}\frac{\sin^{2014}(\pi/2+t)}{\sin^{2014}(\pi/2+t)+\cos^{2014}(\pi/2+t)}dt = \int\limits_{0}^{\pi/2}\frac{\cos^{2014}t}{\sin^{2014}t+\cos^{2014}t}dt = \int\limits_{0}^{\pi/2}\frac{\cos^{2014}x}{\sin^{2014}x+\cos^{2014}x}dx$