sich integrieren
to integrate
To become part of a group or society and adapt to its social and cultural circumstances.
Es fiel ihm leicht, sich in das neue Team zu integrieren.
It was easy for him to integrate into the new team.

to integrate
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to integrate
To become part of a group or society and adapt to its social and cultural circumstances.
Es fiel ihm leicht, sich in das neue Team zu integrieren.
It was easy for him to integrate into the new team.

Pay attention to the difference: 'sich integrieren' (to integrate oneself / to integrate) is reflexive and means that a person becomes part of a group. 'Etwas integrieren' (to integrate something) is transitive and means that you insert a part into a whole.
Die neuen Mitarbeiter haben sich schnell ins Team integriert.
The new employees integrated into the team quickly.
The verb 'integrieren' (to integrate) is almost always used with the preposition 'in' followed by the accusative. This expresses what someone or something is integrated into.
Sie hat sich gut in die neue Schulklasse integriert.
She integrated well into the new school class.
You can clearly recognize the mathematical meaning of 'integrieren' (to integrate) from the context. It is used only in mathematics, often together with words like 'Funktion' (function), 'Kurve' (curve), or 'von ... bis' (from ... to).
Können Sie uns zeigen, wie man diese Funktion integriert?
Can you show us how to integrate this function?
to connect / to bandage
'verbinden' and 'integrieren' can both mean that parts come together. 'verbinden' is more general: you can connect or combine things technically, physically, or conceptually. The parts do not necessarily have to become a new whole. 'integrieren' is used when something is deliberately inserted into a larger whole and then belongs to it. It often sounds more planned and somewhat more formal. In short: 'verbinden' emphasizes the connection between parts, while 'integrieren' emphasizes insertion into a system or unit.

Die Firma integriert das neue Programm in ihr altes System.
The company integrates the new program into its old system.
Dieses Kabel verbindet den Laptop mit dem Bildschirm.
This cable connects the laptop to the screen.