Question

Интегралы


∫ dx/(5+4cosx)

Answer 1

Ответ:

$\int\limits \frac{dx}{5 + 4 \cos(x) } \\ \\ \text{тригонометрическая замена:} \\ \\ t = tg( \frac{x}{2} ) \\ \cos(x) = \frac{1 - {t}^{2} }{1 + {t}^{2} } \\ dx = \frac{2dt}{1 + t {}^{2} } \\ \\ \int\limits \frac{2dt}{1 + t {}^{2} } \times \frac{1}{5 + \frac{4(1 - {t}^{2}) }{1 + {t}^{2} } } = \\ = \int\limits \frac{2dt}{1 + t {}^{2} } \times \frac{1 + {t}^{2} }{5(1 + {t}^{2}) + 4(1 - {t}^{2}) } = \\ = \int\limits \frac{2dt}{5 + 5{t}^{2} + 4 - 4 {t}^{2} } = 2\int\limits \frac{dt}{t {}^{2} + 9 } = \\ = 2\int\limits \frac{dt}{ {t}^{2} + {3}^{2} } = \frac{2}{3} arctg (\frac{t}{3} ) + C = \\ = \frac{2}{3} arctg( \frac{tg( \frac{x}{2}) }{3} ) + C$