Your need is not relevant. Mathematics is not something that just works because you want it to. (Unless you have a magic wand, and can use it well.)
When you took out the k*k term from the second hanf of that kernel, it became just a constant, independent of k. Integration is easy in that case.
With the k^2 term in there, it becomes something for which apparently no analytical solution exists. It is trivial to write down such an expression, and they are embarrasingly common. This part:
which can be trivially simplified to
is something that most students can handle. But with the second part in there, as a function now of k, will be apparently more difficult.
So, what can you do? You can write it as a nested numerical integration, something a tool like integral2 can handle. But that will require values of all the unknown variables, thus M. A numerical integration cannot be done with a symbolic parameter in there.
Note that this term:
exp(4120000*k^2 - M/5)
where k varies from 0 to 9.18, will be almost impossible to handle however. Whatever the value of M is, as k varies, the first part of that will be so huge as to be impossble to compute.
k = 9.18;
So that is a number with 150 million decimal digits. I think perhaps you might not understand how large that number is. Double precision arithmetic cannot handle a number larger than
And your exponential is so much more huge than that. We don't know what M is, of course. But hear my words, you will have huge numerical problems in trying to solve this problem