The Brother MFC-J2740DW, Brother MFC-L2740DW and Brother MFC-L2740DWR models are listed in About AirPrint - Apple Support as printers supporting the AirPrint technology. Which model do you have?
An AirPrint printer can be used without an additional app, provided that the printer and the iPhone are connected to the same network (Wi-Fi router).
AirPrint works with wireless (Wi-Fi) or wired (Ethernet) network connections. A Brother printer that has Ethernet (in addition to wireless) could be wired to one of the LAN ports of the Wi-Fi router, if you so wish. With an Ethernet printer connection, some of the problems related to Wi-Fi may be avoided.
Any change if you try the following power-cycle procedure (do not forget the router)?
Power off the Wi-Fi router, the printer, the iPhone, any iPads and Macs, and all other computers/devices. Wait two to three minutes. Power on the router. Wait two to three minutes. Power on the printer. Wait a minute or two. Power on the iPhone. Power on any iPads and Macs. Finally, power on all other computers/devices.