Drop your jaw slightly with relaxed lips. Touch the tongue tip behind the bottom front teeth and arch the top-front toward the roof.

Americans pronounce integrity as ihn-TEH-gruh-tee (/ɪnˈtɛɡrəɾi/). Stress falls on the second syllable — keep everything else short and quick.
Record yourself saying "integrity" and play it back. The mic stays on your device — nothing's uploaded.
4 syllables, 9 sounds. Tap a syllable to jump to its row, then explore each sound's mouth shape and how it's made.
Raise the back of your tongue to touch the soft palate. Add vocal cord vibration, then release.

Curl or bunch your tongue without letting the tip touch the roof of your mouth. Brace the sides of your tongue against your upper back teeth, and round your lips slightly.
Relax your lips, jaw, and tongue completely. Drop your jaw slightly and keep the tongue neutral.
The textbook way isn't wrong — it's just not how anyone actually says it.
Stress falls on the second syllable, not the others. Stretch TEH — keep everything else short and quick.
Don't pronounce the second syllable too fully. The unstressed syllable reduces to a schwa — the lazy "uh" sound — in casual speech.