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Консультация № 151922
26.11.2008, 15:20
0.00 руб.
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1
Помогите пожалуйста, элементарные примеры, но я ничего не помню... Интеграл 1+sin^2x/sinx*dx; ctg^2xdx. Очень бы срочно!
Обсуждение
Неизвестный
26.11.2008, 15:56
общий
это ответ
Здравствуйте, Антонов Дмитрий Валерьевич!
1) Домнoжим числитель и знаменатель на sinx , знаменатель сразу разложим на 1-((cosx)^2) , а в числителе вынесем за знак дифференциала .
INT[1+sin^2x/sinx*dx]=INT[((1+1-((cosx)^2))/(1-((cosx)^2)))*(sinx)*dx]=-INT[((2-((cosx)^2))/(1-((cosx)^2)))*d(cosx)...
Далее получаются 2 простейших интеграла , табличных если сделать замену сosx=u .
-INT[((2-((cosx)^2))/(1-((cosx)^2)))*d(cosx)=-INT[d(cosx)]+INT[(d(cosx))/(((cosx)^2)-1)]=C-cosx+(1/2)*Ln|(cosx-1)/(cosx+1)| .
2) Сведём к2 простейшим интегралам путём разложения : ((ctgx)^2)=(1/((sinx)^2))-1 .
INT[((ctgx)^2)*dx]=INT[dx/((sinx)^2)]-INT[dx]=C-(ctgx)-x .
Без всяких замен пришли к 2 табличным интегралам .
1) Домнoжим числитель и знаменатель на sinx , знаменатель сразу разложим на 1-((cosx)^2) , а в числителе вынесем за знак дифференциала .
INT[1+sin^2x/sinx*dx]=INT[((1+1-((cosx)^2))/(1-((cosx)^2)))*(sinx)*dx]=-INT[((2-((cosx)^2))/(1-((cosx)^2)))*d(cosx)...
Далее получаются 2 простейших интеграла , табличных если сделать замену сosx=u .
-INT[((2-((cosx)^2))/(1-((cosx)^2)))*d(cosx)=-INT[d(cosx)]+INT[(d(cosx))/(((cosx)^2)-1)]=C-cosx+(1/2)*Ln|(cosx-1)/(cosx+1)| .
2) Сведём к2 простейшим интегралам путём разложения : ((ctgx)^2)=(1/((sinx)^2))-1 .
INT[((ctgx)^2)*dx]=INT[dx/((sinx)^2)]-INT[dx]=C-(ctgx)-x .
Без всяких замен пришли к 2 табличным интегралам .
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