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Консультация № 108069
05.11.2007, 09:39
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Добрый дент! Спасибо экспертам за помощь в нахождении предела.
Прошу помочь в решении такого задания:
найти несобственный интеграл
нижний предел 0 ,верхний + бесконечность, под интегралом xdx/sgrt(x^4+1)
Заранее очень признательна за помощь, жду ответ
Прошу помочь в решении такого задания:
найти несобственный интеграл
нижний предел 0 ,верхний + бесконечность, под интегралом xdx/sgrt(x^4+1)
Заранее очень признательна за помощь, жду ответ
Обсуждение
Неизвестный
05.11.2007, 10:37
общий
это ответ
Здравствуйте, Дроздова Елена Владимировна!
intxdx/sqrt(x^4+1)=|t=x^2, dt=2xdx|=int(1/2*dt)/sqrt(t^2+1)=1/2*ln|t+sqrt(t^2+1)|=
=1/2*ln|x^2+sqrt(x^4+1)|
int_0^{oo}xdx/sqrt(x^4+1)=lim_{b->oo}int_0^b xdx/sqrt(x^4+1)=
=lim_{b->oo}(1/2*ln|x^2+sqrt(x^4+1)|)|_0^b=
=lim_{b->oo}(1/2*ln|b^2+sqrt(b^4+1)|)|-1/2*ln|0+sqrt(0+1)|)|)=
=lim_{b->oo}(1/2*ln|b^2+sqrt(b^4+1)|)|)=oo.
Ответ: oo, расходится.
oo-это бесконечность
int_0^{oo} - это интеграл от 0 до бесконечности
lim_{b->oo} - это предел при b стремящемся к бесконечности
|_0^b - это от нуля до b (формула Ньютона-Лейбница)
intxdx/sqrt(x^4+1)=|t=x^2, dt=2xdx|=int(1/2*dt)/sqrt(t^2+1)=1/2*ln|t+sqrt(t^2+1)|=
=1/2*ln|x^2+sqrt(x^4+1)|
int_0^{oo}xdx/sqrt(x^4+1)=lim_{b->oo}int_0^b xdx/sqrt(x^4+1)=
=lim_{b->oo}(1/2*ln|x^2+sqrt(x^4+1)|)|_0^b=
=lim_{b->oo}(1/2*ln|b^2+sqrt(b^4+1)|)|-1/2*ln|0+sqrt(0+1)|)|)=
=lim_{b->oo}(1/2*ln|b^2+sqrt(b^4+1)|)|)=oo.
Ответ: oo, расходится.
oo-это бесконечность
int_0^{oo} - это интеграл от 0 до бесконечности
lim_{b->oo} - это предел при b стремящемся к бесконечности
|_0^b - это от нуля до b (формула Ньютона-Лейбница)
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