3) integrazi in ambele parti

integrala ln(x^2+1)dx = integrala x' * ln(x^2+1)dx =
 = x * ln(x^2+1) - ~ x * (1/x^2+1) * 2x dx =
 = x * ln(x^2+1) - 2 ~ (x^2 + 1 - 1)/(x^2+1) dx =
 = x * ln(x^2+1) - 2 ~ (x^2+1)/(x^2+1) dx + 2 ~ 1/(x^2+1) =
 = x * ln(x^2+1) - 2x + 2 arctg x

pentru integrala 2x*arctg(x)

faci prin parti
 integrala 2x*arctg(x) dx
 Notezi f'(x) = 2x
 f(x) = x^2
 Notezi g(x) = arctg(x)
 g'(x) = 1 / (x^2 + 1)
 integrala f'(x)g(x) dx = f(x)g(x) - integrala f(x)g'(x) dx
 integrala 2xarctg(x dx = x^2*arctg(x) - integrala x^2 / (x^2 + 1) dx

 Simplifici:
 x^2 / (x^2 + 1)
 (x^2 + 1 - 1) / (x^2 + 1)
 (x^2 + 1) / (x^2 + 1) - 1 / (x^2 + 1)
 1 - 1 / (x^2 + 1)

 Inlocuiesti:
 integrala x^2 / (x^2 + 1) dx
 integrala [1 - 1 / (x^2 + 1)] dx
 integrala 1 dx - integrala 1 / (x^2 + 1) dx
 x - arctg(x)

 Put it all together to integrate the function:
 integrala 2xarctg(x) dx
 =x^2arctg(x - integrala x^2 / (x^2 + 1) dx
 =x^2arctg(x - (x - arctg(x) + C
 =x^2arctg(x - x + arctg(x + C
 =x^2arctg(x + arctg(x) - x + C