I have read in:
that Bitlocker does not provide data integrity since it takes more space in the disk. The PDF says also that drive encryption systems do not offer integrity of the stored data, implementing the so-called poor man authentication:
"The best solution is to use poor-man’s authentication: encrypt the data and trust to the fact that changes in the ciphertext do not translate to semantically sensible changes to the plaintext. For example, an attacker can change the ciphertext of an executable, but if the new plaintext is effectively random we can hope that there is a far higher chance that the changes will crash the machine or application rather than doing something the attacker wants."
Does LUKS offer any integrity checks on users data?
As of Arch's wiki (https://wiki.archlinux.org/index.php/Dm-crypt/Specialties):
"mkinitcpio-chkcryptoboot is a mkinitcpio hook that performs integrity checks during early-userspace and advises the user not to enter his root partition password if the system appears to have been compromised. Security is achieved through an encrypted /boot partition, which is unlocked using GRUB's cryptodisk.mod module, and a root filesystem partition, which is encrypted with a password different from the former. This way, the initramfs and kernel are secured against offline tampering, and the root partition can remain secure even if the /boot partition password is entered on a compromised machine (provided that the chkcryptoboot hook detects the compromise, and is not itself compromised at run-time)."