The function
... f(x) = (x+2)/(x-1) = 1 + 3/(x-1)
is symmetrical about the line y=x, hence is its own inverse.
We can evaluate the desired derivative directly.
... f'(x) -3/(x-1)²
so f'(2) is
... f'(2) = -3/(2-1)²

Simplify the integrands by polynomial division.


Now computing the integrals is trivial.
5.

where we use the power rule,

and a substitution to integrate the last term,

8.

using the same approach as above.
I believe it’s the third option