IMS 2010 Problem 1

Let $0. Prove that $\int\limits_{a}^{b}{({{x}^{2}}+1){{e}^{-{{x}^{2}}}}dx\ge {{e}^{-{{a}^{2}}}}-{{e}^{-{{b}^{2}}}}}$

Solution

since $0, we get $({{x}^{2}}+1){{e}^{-{{x}^{2}}}}\ge 2x{{e}^{-{{x}^{2}}}}$(AM-GM)

then $\int\limits_{a}^{b}{({{x}^{2}}+1){{e}^{-{{x}^{2}}}}\ge \int\limits_{a}^{b}{2x{{e}^{-{{x}^{2}}}}dx}}$

Integral by part for $\int\limits_{a}^{b}{2x{{e}^{-{{x}^{2}}}}dx}=-{{e}^{-{{x}^{2}}}}+c$

therefore $\int\limits_{a}^{b}{({{x}^{2}}+1){{e}^{-{{x}^{2}}}}\ge \int\limits_{a}^{b}{2x{{e}^{-{{x}^{2}}}}dx}}={{e}^{-{{a}^{2}}}}-{{e}^{-{{b}^{2}}}}$

Thank you 😀

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