Answer
$-x e^{-x} -e^{-x}+2 e^{x+1}+C$
Work Step by Step
We will solve the given integral by using integrate-by-parts formula such as: $\int udv=uv-\int v du$
$\displaystyle \int (x e^{-x}+2e^{-x+1}) \ dx=\int xe^{-x} dx +2 \int e^{-x+1} \ dx$
or, $=x (-e^{-x})-\int (-e^{-x}) \ dx+2 \int e^{-x+1} \ dx$
or, $=-x e^{-x} -e^{-x}+2 \int e^{-x+1} \ dx+C$
or, $=-x e^{-x} -e^{-x}+2 e^{x+1}+C$
