Question Number 144291 by SOMEDAVONG last updated on 24/Jun/21
Answered by som(math1967) last updated on 24/Jun/21
![I=∫_0 ^(2021) (((2021−x)^(2021) )/((2021−x)^(2021) +(2021−2021+x)^(2021) ))dx 2I=∫_0 ^(2021) ((x^(2021) +(2021−x)^(2021) )/(x^(2021) +(2021−x)^(2021) ))dx I=(1/2)∫_0 ^(2021) dx=(1/2)[x]_0 ^(2021) =((2021)/2) ∫_0 ^a f(x)dx=∫_0 ^a f(a−x)dx](https://www.tinkutara.com/question/Q144296.png)
$${I}=\int_{\mathrm{0}} ^{\mathrm{2021}} \frac{\left(\mathrm{2021}−{x}\right)^{\mathrm{2021}} }{\left(\mathrm{2021}−{x}\right)^{\mathrm{2021}} +\left(\mathrm{2021}−\mathrm{2021}+{x}\right)^{\mathrm{2021}} }{dx} \\ $$$$\mathrm{2}{I}=\int_{\mathrm{0}} ^{\mathrm{2021}} \frac{{x}^{\mathrm{2021}} +\left(\mathrm{2021}−{x}\right)^{\mathrm{2021}} }{{x}^{\mathrm{2021}} +\left(\mathrm{2021}−{x}\right)^{\mathrm{2021}} }{dx} \\ $$$${I}=\frac{\mathrm{1}}{\mathrm{2}}\int_{\mathrm{0}} ^{\mathrm{2021}} {dx}=\frac{\mathrm{1}}{\mathrm{2}}\left[{x}\right]_{\mathrm{0}} ^{\mathrm{2021}} =\frac{\mathrm{2021}}{\mathrm{2}} \\ $$$$\int_{\mathrm{0}} ^{\boldsymbol{{a}}} \boldsymbol{{f}}\left(\boldsymbol{{x}}\right)\boldsymbol{{dx}}=\int_{\mathrm{0}} ^{{a}} \boldsymbol{{f}}\left(\boldsymbol{{a}}−\boldsymbol{{x}}\right)\boldsymbol{{dx}} \\ $$