Задача №1424

\[ \int\frac{x^2e^xdx}{(x+2)^2} =\left[\begin{aligned}& u=x^2e^x;\;du=(x^2e^x+2xe^x)dx=xe^x(x+2)dx.\\& dv=\frac{dx}{(x+2)^2};\;v=-\frac{1}{x+2}.\end{aligned}\right] =-\frac{x^2e^x}{x+2}+\int{xe^x}dx=\\ =\left[\begin{aligned}& u=x;\;du=dx.\\& dv=e^xdx;\;v=e^x.\end{aligned}\right] =-\frac{x^2e^x}{x+2}+xe^x-\int{e^x}dx =-\frac{x^2e^x}{x+2}+xe^x-e^x+C =\frac{e^x(x-2)}{x+2}+C \]