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Консультация № 108834
10.11.2007, 17:00
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Мне нужно проверить правильно ли значение определенного интеграла integral (ot 0 do 1) x^2/(4*x^6+1)dx. У меня получилось 10,57.
А решая методом прямоугольников или трапеций у меня получается 0,18 и 0,15. Как так.
А решая методом прямоугольников или трапеций у меня получается 0,18 и 0,15. Как так.
Обсуждение
Неизвестный
10.11.2007, 17:56
общий
это ответ
Здравствуйте, lyalya!
Точное значение этого интеграла arctg(2)/6 ≈ 0.185.
∫x²dx/(4x<sup>6</sup>+1) = {замена: t = 2x³, dt = 6x²dx, x²dx = dt/6} = 1/6 * ∫dt/(t²+1) = 1/6 * arctg(t) = 1/6 * arctg(2x³).
∫<sub>0</sub><sup>1</sup>x²dx/(4x<sup>6</sup>+1) = 1/6 * arctg(2x³)|<sub>0</sub><sup>1</sup> = 1/6 * (arctg(2) – arctg(0)) = 1/6 * arctg(2).
Точное значение этого интеграла arctg(2)/6 ≈ 0.185.
∫x²dx/(4x<sup>6</sup>+1) = {замена: t = 2x³, dt = 6x²dx, x²dx = dt/6} = 1/6 * ∫dt/(t²+1) = 1/6 * arctg(t) = 1/6 * arctg(2x³).
∫<sub>0</sub><sup>1</sup>x²dx/(4x<sup>6</sup>+1) = 1/6 * arctg(2x³)|<sub>0</sub><sup>1</sup> = 1/6 * (arctg(2) – arctg(0)) = 1/6 * arctg(2).
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