The integument of the frog is always kept moist because:

Correct! Wrong!

The integument of the frog is always kept moist because frogs perform cutaneous respiration.

Cutaneous respiration is a type of respiration where the organism breath through the skin. Due to the amphibious nature, frogs respire through gills (branchial respiration), skin (cutaneous respiration), buccal cavity (buccopharyngeal respiration), and lungs (pulmonary respiration). In frogs, the pulmonary artery divides into two branches, one branch of the pulmonary artery goes to the lungs and the branch of the pulmonary artery goes to the skin and the mucous membrane of the buccopharyngeal cavity.
The tadpole larva possesses external gills to breathe inside the water. Therefore, the tadpole larva breath predominantly by gills or through branchial respiration. Upon metamorphosis, the larva loses their external gills and develops less specialized lungs. Unlike humans, who possess a rib cage, diaphragm, and muscles for breathing, frogs use the nostril and mouth cavity to inhale and exhale air. When the adult frogs are submerged in water, they respire predominantly through the skin.
The skin of frogs is composed of thin, water/gas-permeable membranous skin. The skin is well vascularized with a large network of blood vessels and capillaries. The skin directly allows the exchange of respiratory gases by the process called diffusion, where respiratory gases are exchanged according to their concentration gradients between. Outside water, mucus glands present in the skin of frog produce mucus, which keeps the skin moist and enable the skin to respire even on dry land.