Font U/esint/m/n/12=esint10 at 12.0pt not loadable: Metric (TFM) file not found

The .tfm is not in the distribution nor on CTAN. You have to create it and put it in the relevant place in your system.

Here is how to do that:

1) run the command prompt in the directory that contains esint10.mf; with MiKTeX, it is …\MiKTeX 2.9\fonts\source\public\esint.

2) from the command prompt, run

mf \mode=localfont; input esint10.mf

You will get these files: esint10.600gf, esint10.log and esint10.tfm. Delete the first two and

3) Create an esint directory in

C:\Users\Your_Name\AppData\Local\MiKTeX\2.9\fonts\tfm\public\ 

and move esint10.tfm into that directory.

Alternatively, if you want all users to have access to this font, put esint10.tfm in a local TeXMF root, say LocalTeXMF\fonts\tfm\public\. If this local root doesn't exist (MiKTeX doesn't define it by default, contrary to TeX Live), you can create one with MiKTeX Settings, Roots tab.

4) Finally refresh the file name data base (FNDB).