let f(x) = ∫_0 ^(π/2) ((cosθ)/(1+xsinθ))dθ
1) determine a explicit form of f(x)
2) calculate ∫_0 ^(π/2) ((sin(2θ))/((1+xsinθ)^2 ))dθ
3) find the values of ∫_0 ^(π/2) ((cosθ)/(1+2cosθ))dθ and ∫_0 ^(π/2) ((sin(2θ))/((1+3sinθ)^2 ))dθ .
let 0<x<1 and Γ(x) =∫_0 ^∞ t^(x−1) e^(−t) dt
1) prove that Γ(x).Γ(1−x) =(π/(sin(πx))) (compliments formulae)
2) calculate Γ(n) and Γ(n+(1/2)) with n from N.