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# (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.

## a. $1,1,2,6$b. $\int_{0}^{\infty} x^{n} e^{-x} \mathrm{d} x=n !$c. $n !$

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##### Kristen K.

University of Michigan - Ann Arbor

##### Michael J.

Idaho State University

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So this is equal to axe. Negative X times ninety bucks from zero to a minus. Into girl from zero to a negative into ninety five. Yes, this is Echo Two Negative, Eh? How's it connected? Us? The integral from zero to a into negative X. Yes. So what? Hey, goes to zero. So what? It goes to infinity? This part goes to one. This is a case on busy with zero. So this issue could want on the one day goes to infinity A Lansky, to make the way goes to zero here we can use Lopez has Ross. This is the tortures and name it. It goes to infinity and over to make a This is the limit. Want over negative too Next and goes to infinity. This's going to therapy. So the result is you want? What then is the co two two? This is a limit. It goes to infinity into girl from zero to a X squared. You two make two jacks Jacks here. We also use, uh my third of integration of cars. This is you called too. This definite into heroin. 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My parts look as it's definite Integral. This's integral from zero to a thanks to kill negative The connective you make you, Max. This is equal to negative X Q You two make you wax from there, eh? Minus auntie girl from zero to a negative eaten ninety fights three Explain the axe. This is equal to negative is to kill you. He'd make you pay wass three terms router hero from integral from there to a square to make you vice. Yes, on from the case. And he's got to What a goes to infinity This is it goes two to three times to this it goes to thiss on DH here This is also goes to zero you know, they also use NOPD through the readout is six this is a three hams two so we can gas weaken guys. So here we have. And if any secret to zero with result is one and if we could want result is also one and we got to with this out is equal to to end. We put three this is six. This's I read hands too And this is two times one wakened guys, this guy's for general. Go and speak to some K. And here it is. 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.

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