Помогите, пожалуйста, вычислить интеграл
int (0 pi/2) dx/(5 - 3 * cos x)

int (0 pi/2) dx/(5 - 3 * cos x) =
= | t = tg (x/2); x = 2 * arctg t; cos x = (1 - t^2)/(1 + t^2); dx = 2/(1 + t^2) dt | =
= int (0 1) (2/(1 + t^2))/(5 - 3 * (1 - t^2)/(1 + t^2)) dt =
= int (0 1) (2/(1 + t^2))/(5 - (3 - 3 * t^2)/(1 + t^2)) dt =
= int (0 1) (2/(1 + t^2))/((5 * (1 + t^2) - (3 - 3 * t^2))/(1 + t^2)) dt =
= int (0 1) (2/(1 + t^2))/((5 + 5 * t^2 - 3 + 3 * t^2)/(1 + t^2)) dt =
= int (0 1) (2/(1 + t^2))/((2 + 8 * t^2)/(1 + t^2)) dt =
= int (0 1) 2/(2 + 8 * t^2) dt = int (0 1) 1/(1 + 4 * t^2) dt =
= int (0 1) 1/(1 + (2 * t)^2) dt = | 2 * t = u; t = u/2; dt = 1/2 du | =
= int (0 2) 1/(1 + u^2) * 1/2 du = 1/2 * int (0 2) 1/(1 + u^2) du =
= 1/2 * (arctg u)_{0}^{2} = 1/2 * (arctg 2 - arctg 0) = 1/2 * arctg 2.

venja! Спасибо вам!