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Консультация № 176725
17.02.2010, 16:57
41.45 руб.
0
2
2
Здравствуйте решите пожалуйста задачу желательно по подробней(хотя бы в 3 действия).
Исследовать сходимость интеграла
Исследовать сходимость интеграла
Обсуждение
17.02.2010, 17:27
общий
это ответ
Здравствуйте, kapezc.
Воспользуемся признаком сравнения: если подынтегральная функция непрерывна на [0,+[$8734$]) и эквивалентна
степенной 1/xn, то при n>1 интеграл сходится, а при n[$8804$]1 расходится.
Подынтегральная функция
f(x)=1/(1+x4)
непрерывна имеет степенную асимптотику 1/x4 (с показателем 4), так как
f(x)/(1/x4)=x4/(1+x4)=1/((1/x4)+1) --> 1 при x-->[$8734$]
Поэтому рассматриваемый интеграл сходится.
Воспользуемся признаком сравнения: если подынтегральная функция непрерывна на [0,+[$8734$]) и эквивалентна
степенной 1/xn, то при n>1 интеграл сходится, а при n[$8804$]1 расходится.
Подынтегральная функция
f(x)=1/(1+x4)
непрерывна имеет степенную асимптотику 1/x4 (с показателем 4), так как
f(x)/(1/x4)=x4/(1+x4)=1/((1/x4)+1) --> 1 при x-->[$8734$]
Поэтому рассматриваемый интеграл сходится.
5
Неизвестный
18.02.2010, 13:08
общий
это ответ
Здравствуйте, kapezc!
Т. К. (1/x^4) > (1/(1+x^4)) , причем при сходимости INT 1->8 [dx/x^4] следует сходимость INT 0->8 [dx/(1+x^4)],
INT 1->8 [dx/x^4] = lim b->8 INT 1->b [dx/x^4] = lim b->8 (-3*x^(-3)) | b по 1 =
lim b->8 (-3*b^(-3) + -3*1^(-3)) = 3 =>
INT 1->8 [dx/x^4] сходится => INT 0->8 [dx/(1+x^4)] сходится
Приложение:
http://window.edu.ru/window_catalog/pdf2txt?p_id=23379&p_page=4
Т. К. (1/x^4) > (1/(1+x^4)) , причем при сходимости INT 1->8 [dx/x^4] следует сходимость INT 0->8 [dx/(1+x^4)],
INT 1->8 [dx/x^4] = lim b->8 INT 1->b [dx/x^4] = lim b->8 (-3*x^(-3)) | b по 1 =
lim b->8 (-3*b^(-3) + -3*1^(-3)) = 3 =>
INT 1->8 [dx/x^4] сходится => INT 0->8 [dx/(1+x^4)] сходится
Приложение:
http://window.edu.ru/window_catalog/pdf2txt?p_id=23379&p_page=4
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