Question

Найти интеграл
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Answer 1

$\int_{0}^{ \frac{\pi}{w} } \sin^{2} (wx + φ_{0}) dx = \\ t = wx + φ_{0}, \: dx = \frac{1}{w} dt \\x = 0: \: t = φ_{0}, \: x = \frac{\pi}{w} : \: t = φ_{0} + \pi \\ = \frac{1}{w} \int _{φ_{0}}^{φ_{0} + \pi} \sin^{2} (t) dt = \frac{1}{w} \int _{φ_{0}}^{φ_{0} + \pi} \frac{1 - \cos(2t) }{2} dt = \\ = \frac{1}{w} \int _{φ_{0}}^{φ_{0} + \pi} \frac{1}{2} dt - \frac{1}{w} \int _{φ_{0}}^{φ_{0} + \pi} \frac{ \cos(2t) }{2} dt = \frac{t}{2w} - \\ - \frac{1}{w} \times \frac{ \sin(2t) }{4} = \frac{2t - \sin(2t) }{4w}|_{φ_{0}}^{φ_{0} + \pi} = \frac{2φ_{0} + 2\pi - \sin(2φ_{0}) }{4w} - \\ - \frac{2φ_{0} - \sin(2φ_{0}) }{4w} = \frac{2φ_{0} + 2\pi - \sin(2φ_{0}) - 2φ_{0} + \sin(2φ_{0}) }{4w} = \\ = \frac{\pi}{2w}$