💬 👋 We’re always here. Join our Discord to connect with other students 24/7, any time, night or day.Join Here!



Numerade Educator



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 $.


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


You must be signed in to discuss.

Video Transcript

for a problem is used integration my parts to show that into girl off axe. Jax is ableto access wax miles into neural stamps from my Xbox. But this problem just a lot. What use? They caught you off box. Big problem. Yes, we could one. Then you promise the court too? From LAX, I think. Use the integration by parts we have integral off ex Jax. It's the control. After bucks, times X minus and we're off. Uh uh. Problem Lux. Pam's Lex Dex. I'm approved this equation. Or be the problem is if half Andrea Rhys functions on DH as promised, continuous proof that into girl from A to B after Lex Jax is he could be half the man I f a minus the integral I'm afraid to have bee. Gee, why you? Why so here we have a hint is part ay on make substitution. Why's the cultural half flux? So if we use substitution, why you caught your rifle marks? Then we have axe. Is the cultural tio fly said the ox. It's the control. Teo. Prom. Why the Why now? The integral to be, uh, likes yaks. It's the culture in general from a to be. And why you? Why? Well, why And then they use said, resulting in part to pay. And we have this into girl thiss one. It's the culture. Why? Why? From a Teo be minus and two girl from half a tohave B You know why Why we can taik what I have to be You have to be. This fund is the cultural B She have a B minus and ham's g half things half on Ji Hye humorous functions Jeff is being this speed has Bean and Jeff is a so this is eight times away So improved Poppy A scene and the case Weir half Andrea Positive functions on DH Hey, it's green. It's up a Security bureau job there, Graham to give a geometric to petition of part being No, we can jog a graph here If this is a function for walks and from eh You we have this is awful and b then ante girl off works from a to B is this part area of this part This part This's integral from end to be half off a fax DX and and to grow from halfway have be Joe. I see why is area us this part Now be tamps. Have b is the rectangle. This rectangle and a comes halfway is this rectangle so we can see big ref Tango minus a small rectangle and minus Ariel. This part is just to area with this pot pipe wants is just to into girl Remained to be af expects. This is C What? The use party too. You want it into girl from one to e out in next. Jax. Moving now. Why couldn't you, Alan, Relax. Interest functions of our necks. Is he too? Axe This's axe in musical too. It's Why then? Integral? I'm want to e Alan Bax. Jax Is he conscious? You times found Minus one times one minus one. It's equal to zero. Enough e you see one. So why is it why you want? This is a call to minus zero minus and to grow you two. Why is it why from zero to one? The answer is why