Trust In Me was a song sung by Kaa, who was using his hypnotic powers to put Mowgli into a trance while he sang (so Kaa would get some breakfast afterwards)
The Sherman Brothers
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The Sherman Brothers, a duo of old man Disney's greatest songwriters, created the song a year before Walt's death. The slow, sinister sounding melody was from another Sherman Brothers song that was written for (but for some reason not used in) Mary Poppins. The brothers concented that the song would "give younger audiences the creeps", so Sterling Holloway (the voice of Kaa) gave his character a hissing, lilting lisp that gave the song a humorous dimension that it would not otherwise have.
Lyrics
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Kaa: [singing] Trussst in me, jussst in me. Shut your eyes, and trussst in me. [speaking] Hold still please. [singing] You can sssleep sssafe and sound, knowing I am around. Ssslip into silent sslumber. Sssail on a sssilver missst. Ssslowly and surely, your senses will ceasse to ressissst. [Mowgli snores] [speaking] You're snoring.
Mowgli: [speaking tiredly] Sorry.
Kaa: [singing] Trust in me, just in me. Shut your eyes, and trussst in me...