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Задать вопрос экспертам

Консультация № 187372

26.05.2013, 13:23

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Орловский Дмитрий

Мастер-Эксперт
319965
1463

26.05.2013, 16:28

общий

это ответ

Здравствуйте, Martin!
Раскладывая подинтегральную функцию, получим знакопеременный ряд
e^(-x⁴)=1-x⁴+(x⁸/2)-(x¹²/6)+...
Для такого ряда погрешность при замене ряда частичной суммой не превосходит модуля первого отброшенного члена. Если ограничиться тремя первыми слагаемыми, то погрешность не превысит величины x¹²/6.
Поэтому после интегрирования мы получим погрешность вычисления не более чем

∫₀^2/3(x¹²/6)dx=(x¹³/78)|₀^2/3=0,00007... . Это обеспечивает требуемую точность.

Таким образом, с нужной точностью, интеграл равен
∫₀^2/3e^(-x⁴)dx=(x-(x⁵/5)+(x⁹/18))|₀^2/3=0,666667-0,026337+0,001445=0,641775

Ответ: 0,642

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