Answer
$e$
Work Step by Step
We will solve the given integral by using integrate-by-parts formula such as: $\int udv=uv-\int v du$
Here, $u=x+1$ and $dv=e^x dx \implies v=e^x$
$\displaystyle \int_0^1 (x+1) e^x \ dx= (x+1) e^x -\int_0^1 e^x dx$
or, $=[(x+1) e^x -e^x]_0^1$
or, $=[(1+1) e^1 -e^1]-[(0+1) e^0 -e^0]$
or, $=(2e-e)-(1-1)$
or, $=e$
