int 2x dx/(x^2 + 1) = int d(x^2)/(x^2 + 1) = | t = x^2 | = int dt/(t + 1) = ln |t + 1| + C = | t = x^2 | =
= ln (x^2 + 1) + C
Тогда
int (-00 +00) 2x dx/(x^2 + 1) = (ln (x^2 + 1))_{-00}^{+00} =
= lim (x->+00) ln (x^2 + 1) - lim (x->-00) ln (x^2 + 1) = | t = -x; x = -t | =
= lim (x->+00) ln (x^2 + 1) - lim (t->+00) ln ((-t)^2 + 1) =
= lim (x->+00) ln (x^2 + 1) - lim (t->+00) ln (t^2 + 1) = 0
Либо по другому:
так как функция 2x/(x^2 + 1) нечетная, то int (-a a) 2x dx/(x^2 + 1) = 0 (в том числе и при a = +00).