I think you can do something like this. Make a file with your integral that takes in k:
function [ F ] = int_fun(k)
fun = @(x)((asin(k*sin(x))-0.7*x).^2);
F=integral(fun,-pi/4,pi/4);
end
Then in your script run:
func = @(k)int_fun(k);
fminunc(func,0);
If you know a better starting point than zero or have constraints on k you can include them at this point.
