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Консультация № 182484
12.03.2011, 10:55
0.00 руб.
0
2
2
Здравствуйте! Прошу помощи в следующем вопросе:
вычислить неопределенный интеграл: dx/(sinx+tgx)
вычислить неопределенный интеграл: dx/(sinx+tgx)
Обсуждение
Неизвестный
12.03.2011, 11:20
общий
это ответ
Здравствуйте, Марина!
∫dx/(sin x+tg x)=∫dx/(sinx+sinx/cosx)=∫cosxdx/(sinxcosx+sinx)
сделаем подстановку t=tg(x/2) sinx=2t/(1+t^2); cosx=(1-t^2)/(1+t^2); dx=2dt/(1+t^2)
∫2(1-t^2)dt/(4t)=1/2∫dt/t-1/2∫tdt=1/2lnt-(t^2)/4+C=ln(tg(x/2))/2-((tg(x/2))^2)/4+C
∫dx/(sin x+tg x)=∫dx/(sinx+sinx/cosx)=∫cosxdx/(sinxcosx+sinx)
сделаем подстановку t=tg(x/2) sinx=2t/(1+t^2); cosx=(1-t^2)/(1+t^2); dx=2dt/(1+t^2)
∫2(1-t^2)dt/(4t)=1/2∫dt/t-1/2∫tdt=1/2lnt-(t^2)/4+C=ln(tg(x/2))/2-((tg(x/2))^2)/4+C
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