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Консультация № 108076
05.11.2007, 10:53
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Добрый день! Пожалуйста, проверьте интеграл от sin^5x
integral sin^5x=integral -cos^6x/6 но если взять производнную то неверно, помогите пожалуйста или подскажите по какой формуле расписывать
очень признательна,жду ответ
integral sin^5x=integral -cos^6x/6 но если взять производнную то неверно, помогите пожалуйста или подскажите по какой формуле расписывать
очень признательна,жду ответ
Обсуждение
Неизвестный
05.11.2007, 11:39
общий
это ответ
Здравствуйте, Дроздова Елена Владимировна!
integral(sin<sup>5</sup>x)dx = integral(sin²x)²sin(x)dx = integral(1-cos²x)²sin(x)dx.
Замена:
t = cos(x), dt = -sin(x)dx.
Получаем
-integral(1-t²)²dt = -integral(1-2t²+t<sup>4</sup>)dt = -t + 2t³/3 – t<sup>5</sup>/5 + C =
делаем обратную замену
= -cos(x) + 2cos³x/3 – cos<sup>5</sup>x/5 + C.
Ответ: -cos(x) + 2cos³x/3 – cos<sup>5</sup>x/5 + C.
integral(sin<sup>5</sup>x)dx = integral(sin²x)²sin(x)dx = integral(1-cos²x)²sin(x)dx.
Замена:
t = cos(x), dt = -sin(x)dx.
Получаем
-integral(1-t²)²dt = -integral(1-2t²+t<sup>4</sup>)dt = -t + 2t³/3 – t<sup>5</sup>/5 + C =
делаем обратную замену
= -cos(x) + 2cos³x/3 – cos<sup>5</sup>x/5 + C.
Ответ: -cos(x) + 2cos³x/3 – cos<sup>5</sup>x/5 + C.
Форма ответа
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