How can one make up for an omitted integral part of the prayer, such as a prostration? And is that done by performing a complete rak'ah and then the prostration of forgetfulness (sujood as-sahw) after the tasleem due to the addition, or is it before the tasleem?

The meaning of catching up on a missed obligatory act from a rak'ah (unit of prayer) is to return to it, perform it, and what comes after it, if the worshipper remembers it before reaching its place in the rak'ah that follows it. In this case, he performs the forgotten pillar, completes his prayer, and prostrates for forgetfulness (sujud (definition) al-sahw). However, if he remembers it after reaching the place of the forgotten act in the rak'ah that follows it, the rak'ah from which the pillar was forgotten becomes invalid, and the rak'ah that follows it takes its place. If the forgotten pillar was from the last rak'ah and he remembers it close to the tasleem (salutation of peace), he should catch up on it by performing it and what comes after it, then make the tasleem and prostrate for forgetfulness. But if a long time passes after the tasleem, the prayer becomes invalid.