Findminimum of Integral

I am trying to find the value of $$u$$ that minimizes the following expression:

q = 1 - Exp[-u/b];
expr = Log[(1/(2 + b)) (1/(1 - q + q^2)) (q +
q^2 + (1 - q)^2 Exp[-u/2])/0.3553];   (*SEC BER equation *)
FindMinimum[Integrate[expr, {b, 1, 10}], {u, 0, 10}]


However, it does not yield any output. What's wrong with the code?

• You may want to replace = by :=. – A.G. Jul 6 at 16:35

You can do something like this.

Clear[expr];
q = 1 - Exp[-u/b];
expr[u_?NumericQ] :=
NIntegrate[
Log[(1/(2 + b)) (1/(1 - q + q^2)) (q + q^2 + (1 - q)^2 Exp[-u/2])/0.3553], {b, 1, 10}];


Now easy to optimize.

FindMinimum[expr[u], {u, 0, 10}]


{-10.5871, {u -> 1.2105}}