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0-1-dx-1-x-6-1-6-pi-3-




Question Number 203544 by laxmanb2111 last updated on 21/Jan/24
∫_0 ^1_  (dx/((1−x^6 )^(1/6) )) =(π/3)
01dx(1x6)16=π3
Answered by DwaipayanShikari last updated on 23/Jan/24
∫_0 ^1 (1/((1−x^n )^(1/n)  ))dx  =(1/n)∫_0 ^1 u^((1/n)−1) (1−u)^(−1/n) du  =(1/n)B((1/n),1−(1/n))  =(1/n)Γ((1/n))Γ(1−(1/n))=(π/(nsin(π/n)))  n=6  =(π/3)
011(1xn)1/ndx=1n01u1n1(1u)1/ndu=1nB(1n,11n)=1nΓ(1n)Γ(11n)=πnsin(π/n)n=6=π3

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