Step By Step Guide - 'The Factorial Function' Detailed Class Note

𝑛! = 𝑛 × (𝑛 − 1) × … × 1
5! = 5 × 4 × 3 × 2 × 1
3! = 3 × 2 × 1
2! = 2 × 1
1! = 1
0! = 1 why they both have the same answer
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To answer that question, we need to understand how the factorial function,is formed.
Starting with this equation??
∞
−𝛼𝑥
1 −𝛼𝑥 ∞ 1 1 1
∫𝑒 𝑑𝑥 = − 𝑒 | = − (𝑒 ∞ − 𝑒 0 ) = − (0 − 1) =
𝛼 0 𝛼 𝛼 𝛼
0
∞ ∞
1
∫ 𝑒 −𝛼𝑥 𝑑𝑥 = Differentiate ∫ 𝑥 2 𝑒 −𝛼𝑥 𝑑𝑥 = (2)(1)𝛼 −3
𝛼
0 both sides 0
∞
𝑑 𝑑 1
∫ 𝑒 −𝛼𝑥 𝑑𝑥 = ∞
𝑑𝛼 𝑑𝛼 𝛼
0 ∫ 𝑥1 𝑒 −𝛼𝑥 𝑑𝑥 = (1)𝛼 −2
∞
𝑑 𝑑 −1 0
∫ 𝑒 −𝛼𝑥 𝑑𝑥 = 𝛼 ∞
𝑑𝛼 𝑑𝛼
0 ∫ 𝑥 2 𝑒 −𝛼𝑥 𝑑𝑥 = (2)(1)2𝛼 −3
∞
𝑑 −𝛼𝑥 0
∫ 𝑒 𝑑𝑥 = −1𝛼 −1−1 ∞
𝑑𝛼
0 ∫ 𝑥 3 𝑒 −𝛼𝑥 𝑑𝑥 = (3)(2)(1)𝛼 −4
∞
0
∫ −𝑥 𝑒 −𝛼𝑥 𝑑𝑥 = −(1)𝛼 −2 ∞

0 ∫ 𝑥 4 𝑒 −𝛼𝑥 𝑑𝑥 = (4)(3)(2)(1)𝛼 −5
∞
0
∫ 𝑥 𝑒 −𝛼𝑥 𝑑𝑥 = (1)𝛼 −2
So we get
0 ∞

∫ 𝑥 𝑛 𝑒 −𝛼𝑥 𝑑𝑥 = 𝑛! 𝛼 −(𝑛+1)
Differentiating both sides
∞ 0
∞
𝑑 𝑑 𝑛!
∫ 𝑥 𝑒 −𝛼𝑥 𝑑𝑥 = (1)𝛼 −2
∫ 𝑥 𝑛 𝑒 −𝛼𝑥 𝑑𝑥 =
𝑑𝛼 𝑑𝛼 𝛼 𝑛+1
0
∞ 0
𝑑 −𝛼𝑥 𝑑
∫𝑥 𝑒 𝑑𝑥 = (1)𝛼 −2
𝑑𝛼 𝑑𝛼
∞
0 When 𝛼 = 1
∫ 𝑥(−𝑥)𝑒 −𝛼𝑥 𝑑𝑥 = −2(1)𝛼 −3 ∞
𝑛!
0 ∫ 𝑥 𝑛 𝑒 −1𝑥 𝑑𝑥 =
∞ 1𝑛+1
0
∫ −𝑥 2 𝑒 −𝛼𝑥 𝑑𝑥 = −(2)(1)𝛼 −3 ∞
This is what is called
0 ∫ 𝑥 𝑛 𝑒 −𝑥 𝑑𝑥 = 𝑛!
0
’The Factorial
Function’