Помогите, пожалуйста, найти интеграл:
int (5 - 3 * x)/(4 - 3 * x^2)^(1/2) dx

int (5 - 3 * x)/(4 - 3 * x^2)^(1/2) dx =
= | x = 2/3^(1/2) * t, t = 3^(1/2) * x/2, dx = 2/3^(1/2) dt | =
= int (5 - 3 * 2/3^(1/2) * t)/(4 - 3 * (2/3^(1/2) * t)^2)^(1/2) * 2/3^(1/2) dt =
= 2/3^(1/2) * int (5 - 2 * 3^(1/2) * t)/(4 - 3 * 4/3 * t^2)^(1/2) dt =
= 2/3^(1/2) * int (5 - 2 * 3^(1/2) * t)/(4 - 4 * t^2)^(1/2) dt =
= 1/3^(1/2) * int (5 - 2 * 3^(1/2) * t)/(1 - t^2)^(1/2) dt =
= 5/3^(1/2) * int dt/(1 - t^2)^(1/2) - int 2 * t/(1 - t^2)^(1/2) dt =
= 5/3^(1/2) * arcsin t + int d(1 - t^2)/(1 - t^2)^(1/2) dt = | u = 1 - t^2 | =
= 5/3^(1/2) * arcsin t + int u^(-1/2) du =
= 5/3^(1/2) * arcsin t + 1/(-1/2 + 1) * u^(-1/2 + 1) + C =
= 5/3^(1/2) * arcsin t + 2 * u^(1/2) + C = | u = 1 - t^2 | =
= 5/3^(1/2) * arcsin t + 2 * (1 - t^2)^(1/2) + C = | t = 3^(1/2) * x/2 | =
= 5/3^(1/2) * arcsin (3^(1/2) * x/2) + 2 * (1 - (3^(1/2) * x/2)^2)^(1/2) + C =
= 5/3^(1/2) * arcsin (3^(1/2) * x/2) + 2 * (1 - 3 * x^2/4)^(1/2) + C =
= 5/3^(1/2) * arcsin (3^(1/2) * x/2) + (4 - 3 * x^2)^(1/2) + C