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 integrated as IHN-tuh-gray-tuhd (/ˈɪntəˌgreɪɾəd/). In "integrated", the "t" right after N is dropped — the tongue skips the T stop and moves directly from the N position to the next sound. This is called the Silent T after N, a small move that separates 'classroom' from 'native'. It comes out as IHN·tuh·GRAY·tuhd. Stress falls on the first syllable — keep everything else short and quick. You'll hear it in sentences like "Wearable technology is becoming more integrated into daily life".
Record yourself saying "integrated" and play it back. The mic stays on your device — nothing's uploaded.
4 syllables, 10 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.
Start with your jaw slightly open and the front of your tongue forward and slightly up. Glide upward, your jaw closes a little more and your tongue arches higher toward the roof of the mouth.
Quickly bounce the front of your tongue against the roof of your mouth. Don't stop the airflow — just a quick tap.

Relax your lips, jaw, and tongue completely. Drop your jaw slightly and keep the tongue neutral.
Touch the tip of your tongue to the roof of your mouth just behind your teeth. Add vocal cord vibration as you release.

The textbook way isn't wrong — it's just not how anyone actually says it.
In "integrated", the "t" right after N is dropped — the tongue skips the T stop and moves directly from the N position to the next sound. /t/ is completely silent — the tongue skips the T stop and moves directly from the N position to the next sound.
In "integrated", the "t" between vowels sounds like a quick "d" — the tongue briefly taps the ridge behind the upper teeth. /t/ or /d/ becomes a quick tap [ɾ] — sounds like a soft D. The tongue briefly taps the ridge behind the upper teeth.
Stress falls on the first syllable, not the others. Stretch IHN — keep everything else short and quick.
Don't pronounce the first syllable too fully. The unstressed syllable reduces to a schwa — the lazy "uh" sound — in casual speech.