This is most likely one of the following XAudio DLLs (list taken from my own PC, date formats may be diferent for you), all present in C:\Windows\System32:
05/03/2008 17:04 489,480 XAudio2_0.dll
30/05/2008 15:19 511,496 XAudio2_1.dll
31/07/2008 11:40 513,544 XAudio2_2.dll
27/10/2008 11:04 518,480 XAudio2_3.dll
16/03/2009 15:18 521,560 XAudio2_4.dll
04/09/2009 18:44 517,960 XAudio2_5.dll
04/02/2010 11:01 530,776 XAudio2_6.dll
02/06/2010 05:55 518,488 XAudio2_7.dll
18/03/2017 21:57 3,584 XAudio2_8.dll
18/03/2017 21:57 574,464 XAudio2_9.dll
When Microsoft still issued the old DirectX SDK they frequently issued new DLLs for certain components with it, bumping the version number, and making them incompatible with any PC that was running an older version of the runtimes. Unfortunately, they also didn't include these updated DLLs with fresh installs of Windows either. The same thing happened with D3DX DLLs and hence the fact that re-installing DirectX appears to fix it. It's not actually a full reinstall of DirectX; all that happens is the missing DLLs are copied over and everything then works.
The good news is that this most likely won't affect your users. Anybody who has run a Direct3D 9 game on their PC has likely already had the DLLs updated. If not the DirectX End-User Runtime Web Installer provides a lighter download that will only grab the missing components.