Problem 60 Hard Difficulty

(a) Evaluate the integral $ \displaystyle \int_0^\infty e^{-x}\ dx $ for $ n = 0, 1, 2 $, and $ 3 $.
(b) Guess the value of $ \displaystyle \int_0^\infty x^n e^{-x}\ dx $ when $ n $ is an arbitrary positive integer.
(c) Prove your guess using mathematical induction.

Answer

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Video Transcript

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Read it. Sure. Okay. Caritas on wine. And should that should do it. Hey, a times, eh? My swan. Times of a dog arms! Who has won? So this's too Hey, Akhtar Ville Potsie! We use induction toe proof. This result Assume what on earth is he going to? OK, the result is K factorial. We need to talk the case and Izzy, go to Cape Plus one for this case. This is a antique girl. There's a limit. It goes to infinity into grow from zero to a ax to a plus one Eat next, you guys? Yes. Here. We also use integration Bypass. This is constant. There are two, eh, Stephanie? The world this is equal to an integral from zero to a Act Two. Okay, last one. Okay, negative. Eat naked. Just physical, too. X two. Okay. Plus one's power. I'm selective to neck you vets from zero to a minus into a girl from the road, eh? Ninety eight ninety vise um, a plus one to place our Yes. This is a call to negative x two k plus one into making survives. Oh, yeah, This is a on DH plus a swan. I'm into girl from zero to a into negative ads. Arms as to is Howard. Yes. Yes, we assume. What is that? Okay, the result is K factorial. So here, when a goes to infinity this's k factorial on k factorial times Keep us one is K Paswan factorial on the first part. This's goes to Cyril Want a costly infinity? You can also use a big ass road proof This is zero. So the result is a class one by induction. So our guys, this is true. We have proved our guests.