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Консультация № 72627
25.01.2007, 19:05
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уважаемые эксперты!
помогите пожалуйста вычислить интеграл:
∫(3х)/(√(x^4+7) dx
помогите пожалуйста вычислить интеграл:
∫(3х)/(√(x^4+7) dx
Обсуждение
Неизвестный
25.01.2007, 19:43
общий
это ответ
Здравствуйте, Dayana!
∫(3х)/(√(x^4+7) dx =3/2∫1/(√(x^4+7) dx^2=|замена x^2=u|=3/2∫1/(√(u^2+7)du du=3/2ln|u+√(u^2+7)|+C
Осталось только обратно заменить u на x.
∫(3х)/(√(x^4+7) dx =3/2∫1/(√(x^4+7) dx^2=|замена x^2=u|=3/2∫1/(√(u^2+7)du du=3/2ln|u+√(u^2+7)|+C
Осталось только обратно заменить u на x.
Неизвестный
25.01.2007, 19:58
общий
это ответ
Здравствуйте, Dayana!
∫(3х)/(√(x^4+7) dx = 3∫x/(√(x^4+7) dx =
= {заносим <b>x</b> под знак дифференциала} =
= (3/2)*∫d(x^2)/(√(x^4+7) = {делаем замену <b>x^2=t</b>}=
= (3/2)*∫dt/(√(t^2+7) = {а это стандартная формула}=
= (3/2)*arcsinh(t/√7) = {делаем обратную замену}=
= (3/2)*arcsinh((x^2)/√7)
<i><b>arcsinh</b> - ареа-синус (арксинус гиперболический).</i>
Good Luck!!!
∫(3х)/(√(x^4+7) dx = 3∫x/(√(x^4+7) dx =
= {заносим <b>x</b> под знак дифференциала} =
= (3/2)*∫d(x^2)/(√(x^4+7) = {делаем замену <b>x^2=t</b>}=
= (3/2)*∫dt/(√(t^2+7) = {а это стандартная формула}=
= (3/2)*arcsinh(t/√7) = {делаем обратную замену}=
= (3/2)*arcsinh((x^2)/√7)
<i><b>arcsinh</b> - ареа-синус (арксинус гиперболический).</i>
Good Luck!!!
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