int (x^2 + x) * e^(-3x) dx = int (x^2 + x) d(-1/3 * e^(-3x)) =
= -1/3 * int (x^2 + x) d(e^(-3x)) =
= -1/3 * (x^2 + x) * e^(-3x) + 1/3 * int e^(-3x) d(x^2 + x) =
= -1/3 * (x^2 + x) * e^(-3x) + 1/3 * int (2x + 1) * e^(-3x) dx =
= -1/3 * (x^2 + x) * e^(-3x) + 1/3 * int (2x + 1) d(-1/3 * e^(-3x)) =
= -1/3 * (x^2 + x) * e^(-3x) - 1/9 * int (2x + 1) d(e^(-3x)) =
= -1/3 * (x^2 + x) * e^(-3x) - 1/9 * (2x + 1) * e^(-3x) + 1/9 * int e^(-3x) d(2x + 1) =
= -1/3 * (x^2 + x) * e^(-3x) - 1/9 * (2x + 1) * e^(-3x) + 2/9 * int e^(-3x) dx =
= -1/3 * (x^2 + x) * e^(-3x) - 1/9 * (2x + 1) * e^(-3x) + 2/9 * (-1/3) * e^(-3x) + C =
= -1/3 * (x^2 + x) * e^(-3x) - 1/9 * (2x + 1) * e^(-3x) - 2/27 * e^(-3x) + C =
= -1/3 * (x^2 + x + 2/9) * e^(-3x) - 1/9 * (2x + 1) * e^(-3x) + C =
= -1/3 * (x^2 + x + 2/9 + 2/3 * x + 1/3) * e^(-3x) + C =
= -1/3 * (x^2 + 5/3 * x + 5/9) * e^(-3x) + C