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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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