Решите неопределенный интеграл
[tex] \int\limits { \frac{dx}{x(1+ \sqrt[3]{x})^2 } } \, dx [/tex]
[tex] \int\limits { \frac{dx}{x(1+ \sqrt[3]{x})^2 } } \, dx [/tex]
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Кор.куб(x) = t, x = t^3, dx = 3t^2 dtInt (3t^2)/[t^3*(1+t)^2] dt = 3*Int 1/[t*(1+t)^2] dt = 3*Int (A1/t + A2/(1+t) + A3/(1+t)^2)
Метод неопр. коэф-тов
A1/t + A2/(1+t) + A3/(1+t)^2 = [A1*(1+t)^2 + A2*t(1+t) + A3*t]/[t*(1+t)^2] =
= [A1*(1 + 2t + t^2) + A2*t + A2*t^2 + A3*t]/[t*(1+t)^2] =
= [t^2*(A1 + A2) + t*(2A1 + A2 + A3) + A1]/[t*(1+t)^2] = 1/[t*(1+t)^2]
{ A1 + A2 = 0
{ 2A1 + A2 + A3 = 0
{ A1 = 1
A2 = -1, A3 = -1
Int (3t^2)/[t^3*(1+t)^2] dt = 3*Int (1/t - 1/(1+t) - 1/(1+t)^2) dt =
= 3*(ln |t| - ln |1+t| + 1/(1+t)) + C = 3*(ln |t/(1+t)| + 1/(1+t)) + C
Подставь обратно t = кор.куб(x)