let f(x)=∫_0 ^∞ (t^(a−1) /(x+t^n )) dt with 0<a<1 and x>0 and n≥2
1) determine a explicit form of f(x)
2) calculate g(x) =∫_0 ^∞ (t^(a−1) /((x+t^n )^2 )) dt
3) find f^((k)) (x) at form of integrals
4) calculate ∫_0 ^∞ (t^(a−1) /(9+t^2 )) dt and ∫_0 ^∞ (t^(a−1) /((9+t^2 )^2 ))
5) calculate U_n =∫_0 ^∞ (t^((1/n)−1) /(2^n +t^n )) dt and study the convergence of Σ U_n
let A_n =∫_0 ^∞ (x^(a−1) /(1+x^n ))dx with n integr and n≥2 and 0<a<1
1) calculate A_n
2) find the values of ∫_0 ^∞ (x^(a−1) /(1+x^2 ))dx and ∫_0 ^∞ (x^(a−1) /(1+x^3 ))dx
3)calculate ∫_0 ^∞ (dx/((√x)(1+x^4 ))) and ∫_0 ^∞ (dx/((^3 (√x^2 ))(1+x^4 )))
The surnames of 40 students in a class were arranged in
alphabetical order. 16 of the surnames begin with O while
9 of the surnames begin with A. 14 of the letters of the
alphabet do not appear as the first letter of any surname.
(i) What is the probability that the surname of a child picked
at random from the class begins with either A or O
(ii) If more than one surname begins with a letter besides A
and O. How many surnames begins with that letter ?