Question Number 154052 by mathdanisur last updated on 13/Sep/21
$$\mathrm{Prove}\:\mathrm{without}\:\mathrm{any}\:\mathrm{software} \\ $$$$\underset{\:\mathrm{2}β\sqrt{\mathrm{3}}} {\overset{\:\mathrm{1}} {\int}}\:\mathrm{e}^{β\boldsymbol{\mathrm{x}}^{\mathrm{2}} } \:\mathrm{dx}\:<\:\frac{\boldsymbol{\pi}}{\mathrm{6}}\:\:\mathrm{and}\underset{\:\mathrm{1}} {\overset{\:\mathrm{2}+\sqrt{\mathrm{3}}} {\int}}\mathrm{e}^{β\boldsymbol{\mathrm{x}}^{\mathrm{2}} } \:\mathrm{dx}\:<\:\frac{\boldsymbol{\pi}}{\mathrm{6}} \\ $$