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Консультация № 138864
01.06.2008, 02:07
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еще раз прошу помощи!! срочно надо решение этого задания...надеюсь на вас!!
Найти неопределенные интегралы используя для этого интегрирование по частям.
S(x+1)ln(x+1)dx
вы последняя моя надежда(
Найти неопределенные интегралы используя для этого интегрирование по частям.
S(x+1)ln(x+1)dx
вы последняя моя надежда(
Обсуждение
01.06.2008, 02:49
общий
это ответ
Здравствуйте, Knopka star!
Выполним сначала замену переменной: x+1 = t. Тогда dx = dt,
и Int (x+1)ln (x+1)dx = Int t*ln t*dt.
Для нахождения последнего интеграла применим интегрирование по частям. Полагаем u = ln t, dv = tdt. Тогда du = dt/t, v = (1/2)t^2, и
Int t*ln t*dt = (1/2)(ln t)/t^2 - (1/2)*Int (t^2)dt/t = (1/2)(ln t)(t^2) - (1/2)*Int tdt =
= (1/2)(ln t)(t^2) - (1/4)t^2 + C = (t^2)((ln t)/2 - 1/4) + C.
Переходя обратно к первоначальной переменной интегрирования, получаем
Int (x+1)*ln (x+1)*dx = ((x+1)^2)[(ln (x+1))/2 - 1/4] + C.
Ответ: ((x+1)^2)[(ln (x+1))/2 - 1/4] + C.
В принципе, замену переменной можно не делать, а сразу применить интегрирование по частям...
Выполним сначала замену переменной: x+1 = t. Тогда dx = dt,
и Int (x+1)ln (x+1)dx = Int t*ln t*dt.
Для нахождения последнего интеграла применим интегрирование по частям. Полагаем u = ln t, dv = tdt. Тогда du = dt/t, v = (1/2)t^2, и
Int t*ln t*dt = (1/2)(ln t)/t^2 - (1/2)*Int (t^2)dt/t = (1/2)(ln t)(t^2) - (1/2)*Int tdt =
= (1/2)(ln t)(t^2) - (1/4)t^2 + C = (t^2)((ln t)/2 - 1/4) + C.
Переходя обратно к первоначальной переменной интегрирования, получаем
Int (x+1)*ln (x+1)*dx = ((x+1)^2)[(ln (x+1))/2 - 1/4] + C.
Ответ: ((x+1)^2)[(ln (x+1))/2 - 1/4] + C.
В принципе, замену переменной можно не делать, а сразу применить интегрирование по частям...
Об авторе:
Facta loquuntur.
Facta loquuntur.
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