int (0 3) x * (9 - x^2)^(1/2) dx Опции

[princess07]

Помогите, пожалуйста, вычислить интеграл
int (0 3) x * (9 - x^2)^(1/2) dx

[Ярослав_]

int (0 3) x * (9 - x^2)^(1/2) dx = int (0 3) (9 - x^2)^(1/2) d(1/2 * x^2) =
= 1/2 * int (0 3) (9 - x^2)^(1/2) d(x^2) = | x^2 = t | =
= 1/2 * int (0 9) (9 - t)^(1/2) dt =
= 1/2 * (1/(-1) * 1/(1/2 + 1) * (9 - t)^(1/2 + 1))_{0}^{9} =
= -1/2 * 2/3 * ((9 - t)^(3/2))_{0}^{9} =
= -1/3 * ((9 - 9)^(3/2) - (9 - 0)^(3/2)) = -1/3 * (-9^(3/2)) = 1/3 * 27 = 9.

[princess07]

Спасибо (IMG:style_emoticons/default/smile.gif)