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Question Number 144291 by SOMEDAVONG last updated on 24/Jun/21

Answered by som(math1967) last updated on 24/Jun/21

$$I={\int }_0^{2021}\frac{{(2021-x)}^{2021}}{{(2021-x)}^{2021}+{(2021-2021+x)}^{2021}}dx$$$$2I={\int }_0^{2021}\frac{x^{2021}+{(2021-x)}^{2021}}{x^{2021}+{(2021-x)}^{2021}}dx$$$$I=\frac{1}{2}{\int }_0^{2021}dx=\frac{1}{2}{\left[x\right]}_0^{2021}=\frac{2021}{2}$$$${\int }_0^{\mathit{a}}\mathit{f}\left(\mathit{x}\right)\mathit{d}\mathit{x}={\int }_0^a\mathit{f}(\mathit{a}-\mathit{x})\mathit{d}\mathit{x}$$

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