int (1 2) (1 + 2 * x^2)/(x^2 * (1 + x^2)) dx Опции

[BeautyS]

Помогите, пожалуйста, найти интеграл
int (1 2) (1 + 2 * x^2)/(x^2 * (1 + x^2)) dx
Заранее очень благодарна (IMG:style_emoticons/default/smile.gif)

[Lion]

int (1 2) (1 + 2 * x^2)/(x^2 * (1 + x^2)) dx =
= int (1 2) ((1 + x^2) + x^2)/(x^2 * (1 + x^2)) dx =
= int (1 2) (1 + x^2)/(x^2 * (1 + x^2)) dx + int (1 2) x^2/(x^2 * (1 + x^2)) dx =
= int (1 2) dx/x^2 + int (1 2) 1/(1 + x^2) dx =
= (-1/x)_{1}^{2} + (arctg x)_{1}^{2} = (-1/2 + 1/1) + (arctg 2 - arctg 1) =
= -1/2 + 1 + arctg 2 - pi/4 = 1/2 + arctg 2 - pi/4