I believe the first step in analyzing an integral character is literature is to define it. Merriam-Webster.com defines "integral" as:
essential to completeness
In terms of a literary character, I believe these characters are referred to, generally, as major or minor characters.
Dr. L. Kip Wheeler defines characters—
The main character of a work of a fiction is typically called the protagonist; the character against whom the protagonist struggles or contends (if there is one), is the antagonist. If a single secondary character aids the protagonist throughout the narrative, that character is the deuteragonist (the hero's "side-kick").
However, an integral character could be a main character or a minor character. It is more the purpose he (or she) serves that is important.
My sense is that an integral character is an essential character in a story or play. For example, in Romeo and Juliet, the servants present while the Capulets prepare for Juliet's wedding are secondary, supporting characters, but not essential to the development of the plot. Romeo, Juliet and even Friar Lawrence are integral characters, as it is around their actions that the plot evolves, while they are also major characters.
In Hamlet, Horatio is an integral character, as is Fortinbras (who does not even appear until the end of the play), but neither is what I would consider a major character. Horatio provides someone through whom Hamlet speaks to the audience as he tries to work out his indecision over avenging his father's death. Horatio is the one person Hamlet trusts in the castle. Fortinbras serves as a foil. His actual stage role is very small, but knowledge of this character is woven through the story; Fortinbras has a great deal in common with Romeo, also as the son of a dead king, and so we are able to compare the reactions of Hamlet as opposed to those of Fortinbras of Norway.
Some integral characters may serve more of a purpose in the context of when their story or play was written. In Romeo and Juliet, the Nurse is the source of comic relief. In Macbeth, the Porter is the source of comic relief. We may not readily understand their humor, but the Elizabethan audience would have. This might have made them integral in these tragedies to give the audience a "breather."
In To Kill a Mockingbird, the "character" of Tim Johnson is an integral one. Tim Johnson's appearance one summer afternoon allows the reader to learn that Atticus, though he hates guns, is an exceptional marksman. The concept of protecting mockingbirds, as reflected in the novel's title, is also introduced by way of Tim Johnson's presence, a theme that runs throughout the novel.
The distinction of the integral character is not that he/she is a major character, but that his/her role is essential to the development of the plot.