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integral of f(t)/(sqrt(x-t))

Source: VJIMC 2013 2.4

May 30, 2021
calculusintegration

Problem Statement

Let F\mathcal F be the set of all continuous functions f:[0,1]→Rf:[0,1]\to\mathbb R with the property ∣∫0xf(t)x−tdt∣≤1for all x∈(0,1].\left|\int^x_0\frac{f(t)}{\sqrt{x-t}}\text dt\right|\le1\enspace\text{for all }x\in(0,1].Compute sup⁡f∈F∣∫01f(x)dx∣\sup_{f\in\mathcal F}\left|\int^1_0f(x)\text dx\right|.
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