Tinkers' Construct tools don't have an integrity system, so TCTB computes one for them:
Integrity = Material Tier Score + Head/Handle Quality Score + Integrity modifier from installed modules
Material Tier: the highest tier among all materials used in the tool, capped at 4.
Head/Handle Quality: equal to 0.001 × average head durability + average head mining speed + average head attack damage + average handle multiplier, rounded to the nearest integer.
The integrity bonuses of every installed module are then added on top.
TCTB allows a single tool to have multiple identical slots.
When several identical slots all have modules installed:
1 + (N - 1) × 10%.
5 × 1.1 = 5.5 final bonus.