$$\int_{-1}^1 (1-x^2)^n \, dx\ge \frac{1}{\sqrt{n}}$$
So, there is a same question about that. Here is the link; Approximating to the identity by polynomials (Terrance Tao Analysis 2 , 3rd ed , page-73, ex.3.8.2)
In that question, integral is separated from different points. Author says that one part of it is greater than $$\frac{4}{3 \, \sqrt{n}}.$$
How one can prove that?