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Консультация № 178229
06.05.2010, 04:00
42.30 руб.
0
3
1
Тема "Геометрические и механические приложения определенного интеграла"
Задание: Найти площадь фигуры, ограниченной линиями:
y=1/((x*x+1)^2)
x>=0
y=0
Задание: Найти площадь фигуры, ограниченной линиями:
y=1/((x*x+1)^2)
x>=0
y=0
Обсуждение
06.05.2010, 09:26
общий
pikvar:
Похоже у Вас ошибка в условии задачи и вместо y=1/((x*x+1)^2) нужно y=1/(x*(x+1)^2)?
Похоже у Вас ошибка в условии задачи и вместо y=1/((x*x+1)^2) нужно y=1/(x*(x+1)^2)?
Об авторе:
Устав – есть устав! Если ты устав – то отдыхай!
Устав – есть устав! Если ты устав – то отдыхай!
Неизвестный
06.05.2010, 13:18
общий
это ответ
Здравствуйте, pikvar.
Получим несобственный интеграл:
S=[$8747$]0[$8734$]dx/(x2+1)2=[$8747$]0[$8734$](x2+1-x2)/(x2+1)2)dx=[$8747$]0[$8734$]dx/(x2+1)-[$8747$]0[$8734$]x2dx/(x2+1)2
Воспользуемся формулой интегрирования по частям:
[$8747$]udv=u*v-[$8747$]vdu
u=x
dv=xdx/(x2+1)2
v=[$8747$]xdx/(x2+1)2=(1/2)*[$8747$](x2+1)-2d(x2+1)=-(1/2)*1/(x2+1)
[$8747$]udv=[$8747$]x2dx/(x2+1)2=-(1/2)*x/(x2+1)+(1/2)*[$8747$]dx/(x2+1)=-(1/2)*x/(x2+1)+(1/2)*arctg(x)
Получим
[$8747$]dx/(x2+1)2=arctg(x)-(-(1/2)*x/(x2+1)+(1/2)*arctg(x))=1/2*(arctg(x)+x/(x2+1))
S=[$8747$]0[$8734$]dx/(x2+1)2=1/2*(arctg(x)+x/(x2+1))|0[$8734$]=(1/2)*limx->[$8734$](arctg(x)+x/(x2+1))=(1/2)*limx->[$8734$]arctg(x)+(1/2)*limx->[$8734$]x/(x2+1)=(1/2)*Pi/2+(1/2)*0=Pi/4
Получим несобственный интеграл:
S=[$8747$]0[$8734$]dx/(x2+1)2=[$8747$]0[$8734$](x2+1-x2)/(x2+1)2)dx=[$8747$]0[$8734$]dx/(x2+1)-[$8747$]0[$8734$]x2dx/(x2+1)2
Воспользуемся формулой интегрирования по частям:
[$8747$]udv=u*v-[$8747$]vdu
u=x
dv=xdx/(x2+1)2
v=[$8747$]xdx/(x2+1)2=(1/2)*[$8747$](x2+1)-2d(x2+1)=-(1/2)*1/(x2+1)
[$8747$]udv=[$8747$]x2dx/(x2+1)2=-(1/2)*x/(x2+1)+(1/2)*[$8747$]dx/(x2+1)=-(1/2)*x/(x2+1)+(1/2)*arctg(x)
Получим
[$8747$]dx/(x2+1)2=arctg(x)-(-(1/2)*x/(x2+1)+(1/2)*arctg(x))=1/2*(arctg(x)+x/(x2+1))
S=[$8747$]0[$8734$]dx/(x2+1)2=1/2*(arctg(x)+x/(x2+1))|0[$8734$]=(1/2)*limx->[$8734$](arctg(x)+x/(x2+1))=(1/2)*limx->[$8734$]arctg(x)+(1/2)*limx->[$8734$]x/(x2+1)=(1/2)*Pi/2+(1/2)*0=Pi/4
5
Форма ответа
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