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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=∫011−tn1−t2ndtt2n=u=12n∫01t1−2n−t1−n1−udu=12n∫01u12n−1−u12n−121−udu=12nψ(12n+12)−12nψ(12n)

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