# Find an example on a sequence of real-valued functions $(f_n(x))_{n\in\mathbb{N}}$ satisfying the given conditions

I am trying to find an example on a sequence of real-valued functions $(f_n(x))_{n\in\mathbb{N}}$ satisfying the following conditions:

i) $f_n$'s are smooth and have compactly supported.

ii) $\int_{\mathbb{R}}f_n(x)dx=0$ for all $n$.

iii) $\int_{\mathbb{R}}|f_n(x)|dx \ge(\ln n)^{-a}$ for all $n$, for some $a>0$.

I have spent a lot of time to search this sequence, but so far I have still not success. Can someone help me this question? Thank you for helping.

Let $$f(x)=\begin{cases}x e^{1/(x^2-1)}&\text{if |x|<1}\\0&\text{if |x|\ge 1}\end{cases}$$ and $f_n(x)=c_nf(x)$ for suitable $c_n$.