You are integrating, if you collecting small parts to make a whole or part of a whole. On the other hand, if you intergrate ....well... you can never intergrate because "intergrate" is NOT a word.
Integrate means to make something into a whole or a part of the whole. Also, it can be described as when members of different ethnic groups or races become one. They become integrated.
"Members of my extended family integrate immediately during family meetings."
Intergrate is a wrong spelling of the word integrate. It not that commonly used (even as an error).