Solution
= Solution
The <expected value> is zero, so the <variance> is $\mathbb E[X^2]$. Symmetry gives
$$
\operatorname{Var}(X)=\frac2N\int_0^2x^2(1+e^{-x})\,dx.
$$
The polynomial term contributes $8/3$. Two applications of <integration by parts> give
$$
\int x^2e^{-x}\,dx=-(x^2+2x+2)e^{-x}+C,
$$
so its definite value on $[0,2]$ is $2-10e^{-2}$. Therefore
$$
\boxed{\operatorname{Var}(X)=\frac{28/3-20e^{-2}}{6-2e^{-2}}.}
$$