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0-x-u-u-1-f-u-du-n-1-x-n-n-x-f-u-du-Prove-that-




Question Number 145408 by Dwaipayan Shikari last updated on 04/Jul/21
∫_0 ^x ⌊u⌋(⌊u⌋+1)f(u)du=Σ_(n=1) ^(⌊x⌋) n∫_n ^x f(u)du    Prove that
$$\int_{\mathrm{0}} ^{{x}} \lfloor{u}\rfloor\left(\lfloor{u}\rfloor+\mathrm{1}\right){f}\left({u}\right){du}=\underset{{n}=\mathrm{1}} {\overset{\lfloor{x}\rfloor} {\sum}}{n}\int_{{n}} ^{{x}} {f}\left({u}\right){du}\:\: \\ $$$${Prove}\:{that} \\ $$

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