1. int (x^2 + 2) dx/(x^3 + x^2 - 2 * x) = int (x^2 + 2) dx/(x * (x^2 + x - 2)) =
= int (x^2 + 2) dx/(x * (x - 1) * (x + 2))
Разложим подинтегральное выражение на простейшие дроби:
(x^2 + 2)/(x * (x - 1) * (x + 2)) = A/x + B/(x - 1) + C/(x + 2)
Домножим обе части равенства на x * (x - 1) * (x + 2)
x^2 + 2 = A * (x - 1) * (x + 2) + B * x * (x + 2) + C * x * (x - 1)
x = 0 => 2 = -2A => A = -1
x = 1 => 3 = 3B => B = 1
x = -2 => 6 = 6C => C = 1
Тогда
int (x^2 + 2) dx/(x * (x - 1) * (x + 2)) = int (-1/x + 1/(x - 1) + 1/(x + 2)) dx =
= -ln |x| + ln |x - 1| + ln |x + 2| + C
2. int (2 * x^5 - 3 * x^2) dx/(1 + 3 * x^3 - x^6) =
= | t = 1 + 3 * x^3 - x^6; dt = 9 * x^2 - 6 * x^5 | =
= -1/3 * int dt/t = -1/3 * ln |t| + C = | t = 1 + 3 * x^3 - x^6 | =
= -1/3 * ln |1 + 3 * x^3 - x^6| + C