1. Concept 7: The Second Fundamental Theorem of Calculus
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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

2. 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)

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

4. 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

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

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

7. 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

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

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

10. 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

11. 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

12. 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

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

14. 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