Question

Найти интеграл: (1+2х)^2/ x^3 dx

Answer 1

$int{frac{(1+2x)^2}{x^3}}dx=int{frac{1+4x+4x^2}{x^3}}dx=\ =int{left(frac1{x^3}+frac{4}{x^2}+frac4xright)}dx=\ =int{left(x^{-3}+4x^{-2}+4x^{-1}right)}dx=\ =int{x^{-3}}dx+4int{x^{-2}}dx+4int{frac1x}dx=\ =frac{1}{-3+1}x^{-3+1}+4cdotfrac1{-2+1}x^{-2+1}+4ln|x|+C=\ =-frac{1}{2x^2}-frac4x+4ln|x|+C$