Nonono! A1278 MacBook Pros do not have an inverter. These laptops use LED backlighting, which uses DC power. Older non-unibody MacBook Pros manufactured before 2008 use CCFL tubes which use AC power, and therefore need an inverter to convert from the DC output at logic board to AC which the inverter can use. Backlight power goes straight from the logic board to the display panel in your model of MacBook Pro.
LED Backlight runs through the LVDS (display) connector, and operates at a high voltage so is a common point of failure due to corrosion or pin burning when subjected to water damage. You will likely need board repair, but you can try cleaning off any corrosion or burnt-looking pins, especially around the LVDS connector where the backlight voltage runs through.
The fuse on these logic boards usually mediocre at protecting from short-circuiting and over-voltage, and therefore it is likely that the fuse is intact but there is an issue with other parts of the circuit, but you can check for continuity of the fuse by putting a multimeter on resistance mode and checking for ~0Ω by putting a test lead on each of the two sides. Do you know where the backlight fuse is? If not I can point you to the correct location.