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Консультация № 185714
30.03.2012, 19:30
67.69 руб.
0
5
3
Уважаемые эксперты! Пожалуйста, ответьте на вопрос:
Требуется вычислить несколько интегралов.Пожалуйста, помогите их решить.Прошу доходчиво и понятно объяснить метод их решения.
1. int ( dx/(x^2 + x + 10)^2 )
2. int ( (x^2)*(sqrt(x^2 + 1))*dx )
3. int ( sqrt(x^2 + x + 1)*dx )
Очень прошу оформить решение хотя бы 1 из них сегодня!
Заранее большое спасибо!
С уважением,
Иван.
Требуется вычислить несколько интегралов.Пожалуйста, помогите их решить.Прошу доходчиво и понятно объяснить метод их решения.
1. int ( dx/(x^2 + x + 10)^2 )
2. int ( (x^2)*(sqrt(x^2 + 1))*dx )
3. int ( sqrt(x^2 + x + 1)*dx )
Очень прошу оформить решение хотя бы 1 из них сегодня!
Заранее большое спасибо!
С уважением,
Иван.
Обсуждение
30.03.2012, 21:40
общий
это ответ
Здравствуйте, Барс Иван!
в) Имеем
Из выражений (1) и (3) следует, что
или, с учётом выражения (2),
С уважением.
в) Имеем
Из выражений (1) и (3) следует, что
или, с учётом выражения (2),
С уважением.
Об авторе:
Facta loquuntur.
Facta loquuntur.
30.03.2012, 21:57
общий
31.03.2012, 00:14
это ответ
Здравствуйте, Барс Иван!
2.
.
Подкорневое выражение подсказывает воспользоваться гиперболическими косинусами и синусами, для которых верно тождество
,
где
, 
.
Делаем замену
,
.
.
В принципе, можно прямо сейчас выразить всё через экспоненты и проинтегрировать. Это можно чуть облегчить, воспользовавшись тождеством
.
.
Заметим, что
.
Осталось выразить y через x.
Квадратное уравнение относительно ey при фиксированном x. Его положительный корень есть
.
Поэтому
.
Эта функция называется обратным гиперболическим синусом. Подставляя это, получим искомую первообразную:
.
2.
.Подкорневое выражение подсказывает воспользоваться гиперболическими косинусами и синусами, для которых верно тождество
,где
, .Делаем замену
,..В принципе, можно прямо сейчас выразить всё через экспоненты и проинтегрировать. Это можно чуть облегчить, воспользовавшись тождеством
..Заметим, что
.Осталось выразить y через x.
Квадратное уравнение относительно ey при фиксированном x. Его положительный корень есть
.Поэтому
.Эта функция называется обратным гиперболическим синусом. Подставляя это, получим искомую первообразную:
.
31.03.2012, 00:15
общий
Подправил 
Об авторе:
"Если вы заметили, что вы на стороне большинства, —
это верный признак того, что пора меняться." Марк Твен
"Если вы заметили, что вы на стороне большинства, —
это верный признак того, что пора меняться." Марк Твен
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