Assuming[{p, m, [CapitalLambda], k} [Element] Reals, Integrate[k/(2 (k^2 + m^2) (-k + p)), {p, 0, [CapitalLambda]}]]
Mathematica's answer is
ConditionalExpression[(k Log[1 - \[CapitalLambda]/k])/(2 (k^2 + m^2)), \[CapitalLambda] == 0 || (\[CapitalLambda] <= 0 && (k >= 0 || k <= \[CapitalLambda])) || (k >= \[CapitalLambda] && k > 0) || (\[CapitalLambda] >= 0 && k <= 0)]
Question:
Are the different conditions separately true i.e. should the answer be read in terms of "Either \[CapitalLambda] = 0
" OR "[CapitalLambda] < 0 and also k >= 0 and k<=[CapitalLambda]"?