The following numerical integral within a numerical integral:

NIntegrate[ Exp[
-NIntegrate[ 1/(y - (1 - Exp[-(5/6) + (2 y)/3])), 
{y, z, 1}]] , 
{z, -(3/2) ProductLog[-(2/(3 E^(5/6)))], 1}]

gives "NIntegrate::nlim: y = z is not a valid limit of integration" error messages prior to giving an estimate 0.250255.

How might I revise my code to avoid the errors? If the estimate is not valid, please explain why to me. Thank you!


1 Answer 1


To ensure that the inner integral never attempts to evaluate before z has a numeric value, use

int1[z_?NumericQ] :=
 NIntegrate[1/(y - (1 - Exp[-(5/6) + (2 y)/3])), {y, z, 1}]

int2 = NIntegrate[Exp[-int1[z]], 
 {z, -(3/2) ProductLog[-(2/(3 E^(5/6)))], 1}]

(* 0.250255 *)

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