UPDATE 5/4/16: It’s just come to my attention that Apple recently changed the expiration length of free provisioning profiles from 90 days to 7, which renders this sideloading method basically dead. I suggest using the website appishare.com as an alternative to download apps on your device without jailbreaking.
In this method, you’ll need to create a free provisioning profile with Xcode, rewrite the cracked IPA file with iModSign, and then install the app on your device with Xcode. I came across this idea through this video and figured out how to do it myself. This method has been tested on both an iPhone 5S and iPad 3 running iOS 9.3.1, and it works fine.