(a) Use integration by parts to show that (b) If f and g are inverse functions and...

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(a) Use integration by parts to show that

| xf'(x) dx J (x) dx = xf(x) –

(b) If f and g are inverse functions and f' is continuous, prove thatF(x) dx = bf(b) – af(a) – g(v) dy (5 (b) Jf (a)


(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.

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