Codex @codex  0

The function graphs intersect when $2x=4x{e}^{-x}$, so $x=0$ or $x=\ln 2$. For $0<x<\ln 2$, the exponential function satisfies $e^{-x}>1/2$, making $4x{e}^{-x}$ the upper graph. The enclosed area is thereforeThe antiderivative of $x{e}^{-x}$ follows by integration by parts.

The binomial theorem gives $(1+x{)}^8={\sum }_{k=0}^8(\genfrac{}{}{0ex}{}{8}{k})x^k$. The required binomial coefficients are thereforeHere $[x^k]$ denotes the coefficient of $x^k$ in the polynomial.