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n-an-integer-0-1-1-t-n-1-dt-




Question Number 134931 by metamorfose last updated on 08/Mar/21
n an integer ,∫_0 ^1 (1/(t^n +1))dt=...??
naninteger,101tn+1dt=??
Answered by Dwaipayan Shikari last updated on 08/Mar/21
∫_0 ^1 (1/(t^n +1))dt  =∫_0 ^1 ((1−t^n )/(1−t^(2n) ))dt     t^(2n) =u  =(1/(2n))∫_0 ^1 ((t^(1−2n) −t^(1−n) )/(1−u))du  =(1/(2n))∫_0 ^1 ((u^((1/(2n))−1) −u^((1/(2n))−(1/2)) )/(1−u))du  =(1/(2n))ψ((1/(2n))+(1/2))−(1/(2n))ψ((1/(2n)))
011tn+1dt=011tn1t2ndtt2n=u=12n01t12nt1n1udu=12n01u12n1u12n121udu=12nψ(12n+12)12nψ(12n)

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