Let $f$ be a positive $\mathcal C^2$ function defined on $[-1,1]$, with a maximum of $1$ attained only at $0$, and a negative second derivative at $0$ equal to $-\alpha$
Then $$ \int_{-1}^{1}f(x)^n\, \mathrm dx \approx \sqrt{\frac{2\pi}{\alpha n}}$$
I remember that it was proven by approximating the function in a neighborhood of $0$ by a Gaussian function. But I can't remember the name of the approximation.