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Консультация № 198368
24.04.2020, 20:50
0.00 руб.
1
1
1
Здравствуйте! Прошу помощи в следующем вопросе:
5. Найти неопределенный интеграл:
∫▒〖arctg^7 3x〗^ /(1+9x²) dt
5. Найти неопределенный интеграл:
∫▒〖arctg^7 3x〗^ /(1+9x²) dt
Прикрепленные файлы:
Обсуждение
28.04.2020, 06:21
общий
это ответ
Здравствуйте, master87!
Дано : f(x) = arctg7(3·x) / (1 + 9·x2)
Найти неопределенный интеграл [$8747$]f(x)·dx
Решение : Как и в Вашей предыдущей консультаци rfpro.ru/question/198367 , обнаружив, что интеграл [$8747$]arctg7(3·x)·dx / (1 + 9·x2) не удаётся взять с помощью табличных интегралов, используем Метод замены переменной (он хорошо описан в статье "Метод замены переменной в неопределенном интеграле. Примеры решений" Ссылка
Заменяем сложное выражение arctg(3·x) переменной t :
t = arctg(3·x)
Приравниваем производные обоих выражений: t' = [arctg(3·x)]' = 3 / (9·x2 + 1)
Получаем новый дифферециал dt = 3·dx / (9·x2 + 1)
Интегрируем по новой переменной t с помощью лёгкого табличного интеграла:
[$8747$]t7·dt / 3 = (1/3)·[$8747$]t7·dt = (1/3)·(1/8) t8 = t8 / 24
Возвращаемся к переменной x , заменяем t на arctg(3·x)
F(x) = (arctg(3·x))8 / 24 + C , где C = const
Ответ : интеграл равен (arctg(3·x))8 / 24 + C
Проверка обратным дифференцированием в Маткаде успешна. Скриншот прилагаю.
[в Маткаде мне пришлось заменить функцию arctg() на аналог atan()]
Дано : f(x) = arctg7(3·x) / (1 + 9·x2)
Найти неопределенный интеграл [$8747$]f(x)·dx
Решение : Как и в Вашей предыдущей консультаци rfpro.ru/question/198367 , обнаружив, что интеграл [$8747$]arctg7(3·x)·dx / (1 + 9·x2) не удаётся взять с помощью табличных интегралов, используем Метод замены переменной (он хорошо описан в статье "Метод замены переменной в неопределенном интеграле. Примеры решений" Ссылка
Заменяем сложное выражение arctg(3·x) переменной t :
t = arctg(3·x)
Приравниваем производные обоих выражений: t' = [arctg(3·x)]' = 3 / (9·x2 + 1)
Получаем новый дифферециал dt = 3·dx / (9·x2 + 1)
Интегрируем по новой переменной t с помощью лёгкого табличного интеграла:
[$8747$]t7·dt / 3 = (1/3)·[$8747$]t7·dt = (1/3)·(1/8) t8 = t8 / 24
Возвращаемся к переменной x , заменяем t на arctg(3·x)
F(x) = (arctg(3·x))8 / 24 + C , где C = const
Ответ : интеграл равен (arctg(3·x))8 / 24 + C
Проверка обратным дифференцированием в Маткаде успешна. Скриншот прилагаю.
[в Маткаде мне пришлось заменить функцию arctg() на аналог atan()]
5
Спасибо большое!
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