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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.


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


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Video Transcript

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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.