Question

Как решается интеграл: dx/x(1+x)^2 .

Answer 1

1/x(1+x)^2 = A/x + B/(1+x) + C/(1+x)^2
A*(1+x)^2 + B*x*(1+x) + C*x = 1
Ax^2 + 2Ax + A + Bx^2 + Bx + Cx = 1
(^0) --> A = 1
(^1) -> 2A + B + C = 0
(^2) -> A + B = 0
--> A = 1, B = -1, C = -1
1/x(1+x)^2 = 1/x - 1/(1+x) - 1/(1+x)^2

--> ∫ dx/x(1+x)^2 = ∫ dx/x - ∫ dx/(1+x) - ∫ dx/(1+x)^2 = ln|x| - ln|1+x| + 1/(1+x) + C