Place a stepladder under the area. Remove crumbling sections of drywall from the crack with a utility knife. Roughen the surface of the crack with a folded sheet of 120-grit sandpaper. Wipe the sanded area with a microfiber tack cloth to remove any dust clinging to the surface.
Drive pairs of drywall screws spaced 12 to 18 inches apart into the ceiling on either side of the crack. Use a cordless drill/driver and sink the screw heads in just below the surface.
Mix a small amount of drywall joint compound in a plastic bucket with water. Stir the mixture to the consistency of thick cream with a 4-inch-wide drywall knife.
Spread a thin layer of drywall compound over the crack and screw indentations. Place lengths of 2-inch-wide mesh tape over the crack and embed the tape into the compound with one or two passes with the drywall knife. Use a firm touch with the flat face of the knife held at a 45-degree angle. Do not dislodge the tape by overworking the compound; one or two passes is sufficient. Allow the compound to dry.
Sand the surface lightly with 220-grit sandpaper to remove any bumps and to feather in the edges of the repair.
Prepare another small batch of drywall compound to the same consistency as before. Place some compound on a 9-inch-square metal plasterer's hawk fitted with a central handle on the underside; this will hold a workable amount of paste and also assist application by provide a mixing surface and a wiping edge for the drywall knife.
Apply a thin 12-inch-wide layer of compound over the repaired section. Use a 12-inch-wide drywall knife to smooth the surface flat and to feather the sides in so that the paste blends in seamlessly. Allow 20 minutes drying time. Sand lightly. Apply a second layer of compound, allow it to dry and sand it. Add a third layer, if necessary.
Wrap a strip of 220-grit sandpaper around a sanding block. Sand the entire area smooth. Wipe the surface with a microfiber tack cloth.
Cover the floor with a dropcloth. Use a long-handled paint roller to apply a coat of drywall primer to the repaired section. Allow time for the primer to dry.
Paint the ceiling with the finish of your choice.