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Yes, but with a caveat. 5.81 supports the R12i via the "Trimble TIP" (Trimble Installation Package) update. If you have the very latest R12i firmware (v6.10+), you may need version 5.90 or newer for full compatibility. Check Trimble's hardware compatibility matrix before downloading.
To download Trimble Business Center 5.81, you should use official Trimble channels to ensure software integrity and licensing compatibility. trimble business center 5.81 download
Before attempting to download and install TBC 5.81, you must ensure your system meets the minimum requirements and that your license is valid for this specific version. Yes, but with a caveat