let give a>0
1) find the value of F(a) = ∫_0 ^∞ ((lnt)/(t^2 +a^2 ))dt
2) find the value of G(a)=∫_0 ^∞ ((aln(t))/((t^2 +a^2 )^2 ))dt
3) find the value of ∫_0 ^∞ ((ln(t))/((t^2 +3)^2 ))dt
let give the sequence of integrals
J_n =∫_0 ^∞ x^n e^(−(x^2 /2)) dx
1) prove that J_n =(n−1)J_(n−2) ∀n≥2
2) calculate J_(2p) and J_(2p+1) by using factoriels.
3) prove that ∀n≥1 J_n ^2 ≤J_(n−1) . J_(n+1) .
4)prove that ((2^(2p) (p!)^2 )/((2p)!)) (1/(√(2p+1))) ≤J_0 ≤ ((2^(2p) (p!)^2 )/((2p)!)) (1/(√(2p)))
5) find a equivalent of ((2^(2p) (p!)^2 )/((2p)!)) (p→+∞)