# Concept 7: The Second Fundamental Theorem of Calculus

The Second Fundamental Theorem of Calculus
The Second Fundamental Theorem of Calculus

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Concept 7: The Second Fundamental Theorem of Calculus
Info Only The First Fundamental Theorem of Calculus was defined as: 𝑎 𝑏 𝑓(𝑥) 𝑑𝑥 And represented a numerical value where 𝑎 and 𝑏 were fixed values

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Concept 7: The Second Fundamental Theorem of Calculus
The Second Fundamental Theorem of Calculus was defined as: 𝑑 𝑑𝑥 𝑎 𝑥 𝑓 𝑡 𝑑𝑡 =𝑓(𝑥)

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Concept 7: The Second Fundamental Theorem of Calculus
𝑑 𝑑𝑥 𝑎 𝑥 𝑓 𝑡 𝑑𝑡 =𝑓(𝑥) 𝐹′(𝑥)=𝑓(𝑥)

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Example 7A: Putting the MENTAL in Fundamental
Use the Second Fundamental Theorem of Calculus to find 𝐹′(𝑥) 𝐹 𝑥 = −1 𝑥 𝑡 4 +1 𝑑𝑡

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Example 7A: Putting the MENTAL in Fundamental
𝑑 𝑑𝑥 𝑎 𝑥 𝑓 𝑡 𝑑𝑡 =𝑓(𝑥)

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Example 7A: Putting the MENTAL in Fundamental
𝐹 𝑥 = −1 𝑥 𝑡 𝑑𝑡 𝐹′ 𝑥 = 𝑥 4 +1

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It’s spelled “Nun” And you’re not funny Get it? None There is no SLE
Student Led Example 7A And you’re not funny Get it? None There is no SLE

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Example 7B: Using the Second Fundamental Theorem of Calculus
Info Only Or as it should be called: An introduction to “u substitution” And Chain Rule for Integrals

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Example 7B: Using the Second Fundamental Theorem of Calculus
Find 𝐹 ′ 𝑥 Chain Rule 𝐹 ′ 𝑥 = 𝑑𝐹 𝑑𝑢 ⋅ 𝑑𝑢 𝑑𝑥

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Example 7B: Using the Second Fundamental Theorem of Calculus
We need a 𝑢 for this to work Because 𝐹 ′ 𝑥 = 𝑑𝐹 𝑑𝑢 ⋅ 𝑑𝑢 𝑑𝑥 𝑑 𝑑𝑢 𝐹 𝑥 𝑑𝑢 𝑑𝑥 Let 𝑢= 𝑥 2 Chain Rule Definition

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Example 7B: Using the Second Fundamental Theorem of Calculus
𝐹 ′ 𝑥 = 𝑑𝐹 𝑑𝑢 ⋅ 𝑑𝑢 𝑑𝑥 𝑑 𝑑𝑢 0 𝑥 2 sin 𝜃 2 𝑑𝜃 𝑑𝑢 𝑑𝑥 = 𝑑 𝑑𝑢 0 𝑢 sin 𝜃 2 𝑑𝜃 𝑑𝑢 𝑑𝑥

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Example 7B: Using the Second Fundamental Theorem of Calculus
𝐹 ′ 𝑥 = 𝑑𝐹 𝑑𝑢 ⋅ 𝑑𝑢 𝑑𝑥 sin 𝜃 2 ⇒ sin 𝑢 2 𝑑 𝑑𝑢 0 𝑢 sin 𝜃 2 𝑑𝜃 𝑑𝑢 𝑑𝑥 sin 𝑢 2 ⇒ sin 𝑥 2 2 𝑑𝑢 𝑑𝑥 =2𝑥

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Example 7B: Using the Second Fundamental Theorem of Calculus
𝐹 ′ 𝑥 = 𝑑𝐹 𝑑𝑢 ⋅ 𝑑𝑢 𝑑𝑥 ⇒ sin 𝑥 (2𝑥) 2𝑥 sin 𝑥 4

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Example 7B: An Intro to u-substitution
Find the derivative of: 𝐹 𝑥 = 𝜋 2 𝑥 3 cos 𝑡 𝑑𝑡 Using the second fundamental theorem

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