Задача 77669 [m]\int \limits_{\pi}^{2\pi}...

6610e13ca4fded5f0eff603b

5/29/2024 8:21:08 PM

[m]\int \limits_{\pi}^{2\pi} \frac{1-\cos x}{(x - \sin x)^2} dx[/m]

626fab9a354ab65fb2f43abb

5/29/2024 8:58:08 PM

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[m]\int \limits_{\pi}^{2\pi} \frac{1-\cos x}{(x - \sin x)^2} dx[/m]
Замена x - sin x = t; dt = (1 - cos x) dx
t(π) = π - sin π = π - 0 = π; t(2π) = 2π - sin 2π = 2π - 0 = 2π
[m]\int \limits_{\pi}^{2\pi} \frac{1-\cos x}{(x - \sin x)^2} dx = \int \limits_{\pi}^{2\pi} \frac{dt}{t^2} = - \frac{1}{t} |_{\pi}^{2\pi} = -\frac{1}{2\pi} + \frac{1}{\pi} = \frac{1}{2\pi}[/m]