* A lightweight system tray utility to keep your system running * Prevent Windows from turning off or locking your PC * Automatically wakes up your computer when you double click on the executable * No need to install - Buy WinCaffeine Crack - Support developersQ: $\int_0^{2\pi}\frac{\sin^3x}{1+\cos^4x}dx$ I'm doing a simple problem here: $$\int_0^{2\pi}\frac{\sin^3x}{1+\cos^4x}dx$$ I am very happy that after some changes on the integral we are left with: $$\int_0^{2\pi}\frac{1+\cos^4x}{1+\cos^4x}dx=\int_0^{2\pi}\frac{1+\cos^2x}{1+\cos^4x}dx=\int_0^{2\pi}\frac{1+\cos^2x}{1+\cos^2x}dx=\frac{\pi}{2}$$ But how did they found this? I used $\sin^3x=\frac{3\sin^2x\cos x}{2}+\frac{\sin x}{2}$ and I know that $$\int_0^{2\pi}\frac{1+\cos^2x}{1+\cos^2x}dx=\frac{\pi}{2}$$ but how does $\int_0^{2\pi}\frac{\sin x}{1+\cos^4x}dx=\frac{\pi}{2}$? A: You have $$ I=\int_{0}^{2\pi} \frac{\sin^3(x)}{1+\cos^4(x)} dx=\int_{0}^{2\pi} \frac{\cos(x)(1-\cos^2(x))}{1+\cos^2(x)} dx=\int_{0}^{2\pi} \frac{1-\cos^2(x)}{1+\cos^2(x)} dx\\ =\int_{0}^{2\pi} \frac{1-\cos(2x)}{1+\cos(2x)} dx
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