let f(t) =∫_0 ^∞ ((cos^2 (tx))/((x^2 +3)^2 )) dx with t ≥0
1) give a explicit form of f(t)
2) find the value of ∫_0 ^∞ ((xsin(2tx))/((x^2 +3)^2 )) dx
3) give the values of integrals ∫_0 ^∞ (dx/((x^2 +3)^2 )) and ∫_0 ^∞ ((cos^2 (πx))/((x^2 +3)^2 ))dx
4) give the values of integrals ∫_0 ^∞ ((xsin(πx))/((x^2 +3)^2 )) and ∫_0 ^∞ ((xsin(((πx)/2)))/((x^2 +3)^2 )) dx .
let f(λ) =∫_(−∞) ^(+∞) ((sin(λx))/((x^2 +2λx +1)^2 ))dx with ∣λ∣<1
1) find the value of f(λ)
2) calculate ∫_(−∞) ^(+∞) ((sin((x/(2 ))))/((x^2 +x+1)^2 ))dx
3) find A(θ) =∫_(−∞) ^(+∞) ((sin((cosθ)x))/((x^2 +2cosθ x +1)^2 )) that we suppose 0<θ<(π/2)