Question Number 100450 by 175 last updated on 26/Jun/20

Answered by maths mind last updated on 26/Jun/20

∫_0 ^1 (((1−x)^a )/((√x).(√(1−x)).(√(1+x))))dx=f(a)  =∫_0 ^1 x^(−(1/2)) (1−x)^(a−(1/2)) (1+x)^(−(1/2)) dx  ∫_0 ^1 x^((1/2)−1) (1−x)^(a+1−(1/2)−1) .(1−(−1)x)^(−(1/2)) dx=f(a)  f(a)=β((1/2),a+(1/2))._2 F_1 ((1/2),(1/2);a+1;−1)  we can see f′(a)=∫_0 ^1 ((ln(1−x)(1−x)^a )/((√(x.))(√(1−x^2 ))))dx  we want f′(0)  ∂_a β((1/2),a+(1/2))=β((1/2),a+(1/2))(Ψ((1/2))−Ψ(a+1))  too bee continued