int x/(x + 1) dx = int (x + 1 - 1)/(x + 1) dx = int (1 - 1/(x + 1)) dx = x - ln |x + 1| + C

int (0 1) dx/(1 + 2x^2) = |2^(1/2) * x = t; x = t/2^(1/2); dx = dt/2^(1/2) | = 1/2^(1/2) * int (0 2^(1/2)) dt/(1 + t^2) = 1/2^(1/2) * arctg 2^(1/2)