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Prove this integral inequality

Source: 2019 Jozsef Wildt International Math Competition-W. 12

May 18, 2020

integrationinequalitiescalculus

Problem Statement

0 < a < b

then:

\frac{\int \limits^{\frac{a+b}{2}}_{a}\left(\tan^{-1}t\right)dt}{\int \limits_{a}^{b}\left(\tan^{-1}t\right)dt}<\frac{1}{2}