I know this is an old post, but it tops google if you are trying to get your Brother 7840W to work with Airprint.
If you have a MAC, you can use "direct to Internet" and just type in the IP listed on the brother printer (after you set it up), example you use the setup wizard on the printer itself, then look at the settings, it will say IP ex. 192.168.1.5.
Go to Printers in your MAC setting you should be all set.
FOR iPAD/iPHONE etc:
The latest BROTHER PRINTER APP (just look up brother printing), it works really well with the 7840W - Scanning is slow but efficienct, printing is FAST and easy. The app itself is simple, you can copy and past from other apps and basically print directly from any iCloud. You can also use the SCAN to ipad feature, which is nice if you have to keep track of your receipts or want to save pictures given to you in your iCloud.
Hope this helps anyone.