Harold’s Fundamental Theorem of Calculus
“Cheat Sheet”
26 January 2015
Formulas and Rules
Derivation #1
Formulas needed for proof:
Derivation of general formula:
(1) Equivalent Notation
Break integral into two parts
ℎ(𝑥)
𝑦 = 𝑓(𝑥)
∫ 𝑓(𝑡) 𝑑𝑡 =
𝑑𝑦
𝑑𝑓(𝑥)
𝑦 =
= 𝑓 ′ (𝑥) =
𝑑𝑥
𝑑𝑥
′
𝑔(𝑥)
𝑔(𝑥)
𝑎
(3) Change of Bounds
ℎ(𝑥)
𝑎
∫ 𝑓(𝑥) 𝑑𝑥 = − ∫ 𝑓(𝑥) 𝑑𝑥
(4) First Fundamental Theorem of Calculus
𝑎
Using the First Fundamantal Theorem of Calculus
we get
ℎ(𝑥)
∫ 𝑓(𝑡) 𝑑𝑡 = 𝐹(ℎ(𝑥)) − 𝐹(𝑔(𝑥))
𝒃
𝑔(𝑥)
∫ 𝒇(𝒙) 𝒅𝒙 = 𝑭(𝒃) − 𝑭(𝒂)
Take the derivative to get
𝒂
(5a) Second Fundamental Theorem of Calculus
If
ℎ(𝑥)
𝑑
𝑑
𝑑
∫ 𝑓(𝑡) 𝑑𝑡 =
𝐹(ℎ(𝑥)) −
𝐹(𝑔(𝑥))
𝑑𝑥
𝑑𝑥
𝑑𝑥
𝑥
∫ 𝑓(𝑡) 𝑑𝑡 = 𝐹(𝑥)
𝑎
𝑥
𝑑
∫ 𝑓(𝑡) 𝑑𝑡 = 𝐹 ′ (𝑥) = 𝑓(𝑥)
𝑑𝑥
𝑎
(5b) General Formula
𝒉(𝒙)
𝒅
∫ 𝒇(𝒕) 𝒅𝒕 = 𝒇(𝒉(𝒙))𝒉′ (𝒙) − 𝒇(𝒈(𝒙))𝒈′(𝒙)
𝒅𝒙
𝒈(𝒙)
𝑎
𝑔(𝑥)
= ∫ 𝑓(𝑡) 𝑑𝑡 − ∫ 𝑓(𝑡) 𝑑𝑡
𝑎
𝑏
ℎ(𝑥)
∫ 𝑓(𝑡) 𝑑𝑡 = − ∫ 𝑓(𝑡) 𝑑𝑡 + ∫ 𝑓(𝑡) 𝑑𝑡
𝑔(𝑥)
Then
𝑎
𝑔(𝑥)
ℎ(𝑥)
𝑑𝑦
𝑑𝑦 𝑑𝑡
=
∙
𝑑𝑥
𝑑𝑡 𝑑𝑥
𝑎
∫ 𝑓(𝑡) 𝑑𝑡 + ∫ 𝑓(𝑡) 𝑑𝑡
Changing bounds gives
(2) Derivative Chain Rule
𝑏
ℎ(𝑥)
𝑎
𝑔(𝑥)
Using the chain rule we get
ℎ(𝑥)
𝑑
∫ 𝑓(𝑡) 𝑑𝑡 = 𝐹 ′ (ℎ(𝑥))ℎ′ (𝑥) − 𝐹 ′ (𝑔(𝑥))𝑔′(𝑥)
𝑑𝑥
𝑔(𝑥)
Using the Second Fundamantal Theorem of
Calculus we see that 𝐹 ′ (ℎ(𝑥)) = 𝑓(ℎ(𝑥)) and
𝐹 ′ (𝑔(𝑥)) = 𝑓(𝑔(𝑥)).
This completes the proof of the general formula.
Copyright © 2012-2015 by Harold Toomey, WyzAnt Tutor
Example
Solve with the general formula:
Derivation #2
Alternate derivation of the general formula:
𝑥2
𝑑
∫ 𝑒 𝑡 𝑑𝑡
𝑑𝑥
ℎ(𝑥)
𝑑
𝑑
[𝐹(𝑡)]ℎ(𝑥)
∫ 𝑓(𝑡) 𝑑𝑡 =
𝑔(𝑥)
𝑑𝑥
𝑑𝑥
4x
𝑔(𝑥)
ℎ(𝑥)
𝑑
∫ 𝑓(𝑡) 𝑑𝑡 = 𝑓(ℎ(𝑥)) ℎ′ (𝑥) − 𝑓(𝑔(𝑥)) 𝑔′ (𝑥)
𝑑𝑥
𝑔(𝑥)
Here 𝑓(𝑥) = 𝑒 𝑡 , 𝑔(𝑥) = 4𝑥, and ℎ(𝑥) = 𝑥 2
𝑓(ℎ(𝑥)) = 𝑒 𝑥
=
𝑑
𝑑
𝐹(ℎ(𝑥)) −
𝐹(𝑔(𝑥))
𝑑𝑥
𝑑𝑥
= 𝐹 ′ (ℎ(𝑥))
𝑑
𝑑
ℎ(𝑥) − 𝐹 ′ (𝑔(𝑥))
𝑔(𝑥)
𝑑𝑥
𝑑𝑥
2
𝑓(𝑔(𝑥)) = 𝑒 4𝑥
= 𝐹 ′ (ℎ(𝑥)) ℎ′ (𝑥) − 𝐹 ′ (𝑔(𝑥)) 𝑔′ (𝑥)
= 𝑓(ℎ(𝑥)) ℎ′(𝑥) − 𝑓(𝑔(𝑥)) 𝑔′(𝑥)
𝑑 2
𝑥 = 2𝑥
𝑑𝑥
𝑑
𝑔′(𝑥) =
4𝑥 = 4
𝑑𝑥
ℎ′ (𝑥) =
Plug them into the formula and simplify to get
𝑥2
𝑑
∫ 𝑒 𝑡 𝑑𝑡
𝑑𝑥
4x
2
= 𝑒 𝑥 (2𝑥) − 𝑒 4𝑥 (4)
2
= 2𝑥𝑒 𝑥 − 4𝑒 4𝑥
Solve with alternate derivation:
𝑥2
𝑑
∫ 𝑒 𝑡 𝑑𝑡
𝑑𝑥
4x
=
=
d 𝑡 𝑥2
[𝑒 ]4𝑥
dx
d 𝑥 2 d 4𝑥
𝑒 − 𝑒
dx
dx
2
= 𝑒 𝑥 (2𝑥) − 𝑒 4𝑥 (4)
2
= 2𝑥𝑒 𝑥 − 4𝑒 4𝑥
Copyright © 2012-2015 by Harold A. Toomey, WyzAnt Tutor
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Copyright © 2012-2015 by Harold A. Toomey, WyzAnt Tutor
3