Evaluate $\lim \limits_{n \to \infty} n \int_{-1}^{0}(x+e^x)^n dx$. Evaluate $\lim \limits_{n \to \infty} n $$\int_{-1}^{0}(x+e^x)^n  dx$. The answer should be $\frac{1}{2}$. I tried the substitution $x+e^x=u$ and then using the property that $\lim_{n \to \infty } n \int_{-a}^{1} x^n f(x)dx=f(1)$ but I don't know what to do further. If you solve this please do it so that a highschooler like me can understand it, thank you.

Your property says that the limit is
$$
\frac{1}{1+e^{X(1)}},
$$
where $X(u)$ is the solution of $X(u)+e^{X(u)}=u$.  Observe that $e^{X(1)}=1-X(1)$, so your limit is
$$
\frac{1}{2-X(1)}.
$$
It suffices to show that $X(1)=0$.  But this is clear, since $1=X(1)+e^{X(1)}\ge e^{X(1)}\ge 1$, with equality if and only if $X(1)=0$.
