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WZ
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Problem 72 Hard Difficulty

(a) Use integration by parts to show that
$$ \int f(x) dx = xf (x) - \int xf^\prime (x) dx $$
(b) If $ f $ and $ g $ are inverse functions and $ f^\prime $ is continuous, prove that
$$ \int_a^b f(x) dx = bf (b) - af (a) - \int_{f(a)}^{f(b)} g(y) dy $$
[Hint: Use part (a) and make the substitution $ y = f(x) $.]
(c) In the case where $ f $ and $ g $ are positive functions and $ b > a > 0 $, draw a diagram to give a geometric interpretation of part (b).
(d) Use part (b) to evaluate $ \displaystyle \int_1^e \ln x dx $.

Answer

a) Choose $u=f(x), d v=d x,$ find $d u$ and $v,$ then put everything into the integration by parts formula.
b) $\int_{a}^{b} f(x) d x=b f(b)-a f(a)-\int_{f(a)}^{f(b)} g(y) d y$
c) See the image
d) 1

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Video Transcript

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