BUG: read_excel loading some xlsx ints as floats
- [x] closes #46988
- [x] [Tests added and passed](https://pandas.pydata.org/pandas-docs/dev/development/contributing_codebase.html#writing-tests) if fixing a bug or adding a new feature
- [x] All [code checks passed](https://pandas.pydata.org/pandas-docs/dev/development/contributing_codebase.html#pre-commit).
- [x] Added an entry in the latest `doc/source/whatsnew/v1.4.3.rst` file

Thanks @ahawryluk  Owee, I'm MrMeeseeks, Look at me.

There seem to be a conflict, please backport manually. Here are approximate instructions:

1. Checkout backport branch and update it.

```
git checkout 1.4.x
git pull
```

2. Cherry pick the first parent branch of the this PR on top of the older branch:
```
git cherry-pick -x -m1 dea624a0075d595def179b96be27560cf6257438
```

3. You will likely have some merge/cherry-pick conflict here, fix them and commit:

```
git commit -am 'Backport PR #47121: BUG: read_excel loading some xlsx ints as floats'
```

4. Push to a named branch:

```
git push YOURFORK 1.4.x:auto-backport-of-pr-47121-on-1.4.x
```

5. Create a PR against branch 1.4.x, I would have named this PR:

> "Backport PR #47121 on branch 1.4.x (BUG: read_excel loading some xlsx ints as floats)"

And apply the correct labels and milestones.

Congratulations — you did some good work! Hopefully your backport PR will be tested by the continuous integration and merged soon!

Remember to remove the `Still Needs Manual Backport` label once the PR gets merged.

If these instructions are inaccurate, feel free to [suggest an improvement](https://github.com/MeeseeksBox/MeeseeksDev).
                