Identical integral evaluating differently

Question

I am trying to compute an integral but am getting two different answers using two near-identical methods. I wish to know why they differ.

f[x_] := x^2 + 1
g[x_] := 1/(2 π) 1/(x^2 + (1/4))

Simplify[
 Assuming[λ ∈ Reals, 
  Integrate[(
   f[x] g[x])/(λ + I x), {x, -∞, ∞}, 
   PrincipalValue -> True
  ]
 ], 
 λ > 0
]
(* Answer: (5+2λ)/(2+4λ) if λ < 1/4*) 

Assuming[λ > 0, 
 Integrate[
  (f[x] g[x])/(λ + I x), {x, -∞, ∞}, 
  PrincipalValue -> True
 ]
]
(* Answer: (2+λ)/(1+2λ)  if λ <= 1/4*) 

As shown in the code, the two answers are different. I think the latter answer is correct mathematically. However, based on the underlying physics from which this equation arose, the first answer is correct. (I need the integral to yield 2.5 as lambda goes to 0.) I would like to know what causes this mismatch.

Answer 1

If you expand the integrand to ReIm

reim=(f[x] g[x])/(\[Lambda] + I x) // ComplexExpand[ReIm[#]] & // Simplify
(*{(2 (1 + x^2) \[Lambda])/(\[Pi] (1 + 4 x^2)(x^2 + \[Lambda]^2)), -((2 (x + x^3))/(\[Pi] (1 + 4 x^2)(x^2 + \[Lambda]^2)))}*)

integration gives (Mathematica v12.2)

Integrate[reim, {x, -Infinity, Infinity}]

$\left\{\fbox{$\frac{\lambda +2}{2 \lambda +1}\text{ if }\Re(\lambda )>0$},\int_{-\infty }^{\infty } -\frac{2 \left(x^3+x\right)}{\pi \left(4 x^2+1\right) \left(\lambda ^2+x^2\right)} \, dx\right\}$

That means only integration over real part converges.