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Консультация № 159661
05.02.2009, 10:11
0.00 руб.
0
1
1
Здравствуйте.
Помогите пожалуйста решить интегралы.
неопределённый
(sin(x)/(1-cos(x))^2)dx
определённый
сверху 16 снизу 1 [dx/ (sqrt(x) + корень в 4 степени X]
сверху 1 снизу 0 [x^2dx/корень(1-x^3 )]
Спасибо...
Помогите пожалуйста решить интегралы.
неопределённый
(sin(x)/(1-cos(x))^2)dx
определённый
сверху 16 снизу 1 [dx/ (sqrt(x) + корень в 4 степени X]
сверху 1 снизу 0 [x^2dx/корень(1-x^3 )]
Спасибо...
Обсуждение
Неизвестный
05.02.2009, 19:13
общий
это ответ
Здравствуйте, !
1) INT[(sin(x)/(1-cos(x))^2)dx]=INT[d(1-cosx)/((1-cosx)^2)]=C-[1/(1-cosx)] , C=const .
2) Делаем замену : x=y^4 => dx=4*(y^3)*dy , 1<=y<=2 .
сверху 16 снизу 1 [dx/ (sqrt(x) + корень в 4 степени X] = INT{2;1}[4*(y^3)*dy/((y^2)+y)] = 4*INT{2;1}[(y^2)*dy/(y+1)] =
= 4*INT{2;1}[(y-1)*dy] + 4*INT{2;1}[dy/(y+1)] = 2*((y-1)^2)+4*Ln|y+1| = { подставляем новые пределы интегрирования } =
= 2*((2-1)^2)-2*((1-1)^2)+4*(Ln|2+1|-Ln|1+1| = 2*1-2*0+4*(Ln|3|-Ln|2|)=2+4*Ln|3/2| .
3) Как в первом задании вводим весь числитель под знак дифференциала .
сверху 1 снизу 0 [x^2dx/корень(1-x^3 )] = (1/3)*INT[3*(x^2)*dx/sqrt(1-(x^3))] = (1/3)*INTd(1-(x^3))/sqrt(1-(x^3))] =
= (2/3)*sqrt(1-(x^3)) = { не забудем теперь подставить пределы интегрирования вместо х ) =
= (2/3)*(sqrt(1-1)-sqrt(1-0)) = (2/3)*(0-1) = -2/3 .
1) INT[(sin(x)/(1-cos(x))^2)dx]=INT[d(1-cosx)/((1-cosx)^2)]=C-[1/(1-cosx)] , C=const .
2) Делаем замену : x=y^4 => dx=4*(y^3)*dy , 1<=y<=2 .
сверху 16 снизу 1 [dx/ (sqrt(x) + корень в 4 степени X] = INT{2;1}[4*(y^3)*dy/((y^2)+y)] = 4*INT{2;1}[(y^2)*dy/(y+1)] =
= 4*INT{2;1}[(y-1)*dy] + 4*INT{2;1}[dy/(y+1)] = 2*((y-1)^2)+4*Ln|y+1| = { подставляем новые пределы интегрирования } =
= 2*((2-1)^2)-2*((1-1)^2)+4*(Ln|2+1|-Ln|1+1| = 2*1-2*0+4*(Ln|3|-Ln|2|)=2+4*Ln|3/2| .
3) Как в первом задании вводим весь числитель под знак дифференциала .
сверху 1 снизу 0 [x^2dx/корень(1-x^3 )] = (1/3)*INT[3*(x^2)*dx/sqrt(1-(x^3))] = (1/3)*INTd(1-(x^3))/sqrt(1-(x^3))] =
= (2/3)*sqrt(1-(x^3)) = { не забудем теперь подставить пределы интегрирования вместо х ) =
= (2/3)*(sqrt(1-1)-sqrt(1-0)) = (2/3)*(0-1) = -2/3 .
Форма ответа
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