(a) Use integration by parts to show that

(b) If f and g are inverse functions and f' is continuous, prove that

(c) In the case where f and t are positive functions and b > a > 0, draw a diagram to give a geometric interpretation of part (b).
(d) Use part (b) to evaluate ∫^e₁ ln x dx.

| xf'(x) dx J (x) dx = xf(x) F(x) dx = bf(b) af(a) g(v) dy (5 (b) Jf (a)