You have to get to the bottom of why it cracked otherwise any repair you do will crack as well.

We already know why it cracked and fixed that problem.

Cut the bad section out and repair with a patch. Hard to tell exactly how large (or small) this is, so either us a patch kit (found in the paint section of Lowes/Home Depot) or if large you will need to get a sheet of drywall (matching size 3/4", 1/2" etc) cut drywall to fit, screw in, tape edges, mud, sand, sand some more.. prime and paint.

Here one of the DIY kits for holes up to 6"
http://www.lowes.com/pd_345762-52595-DP126_4294858287__?productId=3408480&Ns=p_product_avg_rating|1&pl=1&currentURL=%3FNs%3Dp_product_avg_rating|1&facetInfo=