A new species 2021!

Calculus Level 3

∫ 0 1 1 x 2020 − x 2021 2021 d x = α \huge\mathrm{\int_0^1 \frac{1}{\sqrt[2021]{x^{2020}-x^{2021}}}dx}=\alpha

Find ⌊ α ⌋ ⌊\alpha⌋


The answer is 2021.

Dwaipayan Shikari by Dwaipayan Shikari , 17, India

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1 solution

Dwaipayan Shikari
Dec 30, 2020

∫ 0 1 1 x 2020 − x 2021 2021 d x = ∫ 0 1 x − 2020 2021 ( 1 − x ) − 1 2021 d x \int_0^1 \frac{1}{\sqrt[2021]{x^{2020}-x^{2021}}}dx= \int_0^1 x^{-\frac{2020}{2021}}(1-x)^{-\frac{1}{2021}}dx ⟹ B ( 1 2021 , 2020 2021 ) = Γ ( 1 2021 ) Γ ( 2020 2021 ) Γ ( 1 ) = π sin ⁡ ( π 2021 ) \implies\Beta{(\frac{1}{2021},\frac{2020}{2021})}=\frac{\Gamma{(\frac{1}{2021})}\Gamma{(\frac{2020}{2021})}}{\Gamma{(1)}}= \frac{π}{\sin{(\frac{π}{2021})}}

Which is ≈ 2021.008 ≈2021.008

Answer is ⌊ α ⌋ = 2021 \color{#20A900}\boxed{⌊\alpha⌋ = 2021}

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