Question

Найти интеграл
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Answer 1

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