> For the complete documentation index, see [llms.txt](https://solomons-mathbook.gitbook.io/calculus-basics/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://solomons-mathbook.gitbook.io/calculus-basics/comment-394073989/comment-395305706.md).

# Fundamental Theorem of Calculus (FTC)

> This is somehow dreaded and mind-blowing. But it's the **only** thing to relate the `Differential Calculus` & `Integral Calculus`.

![image](https://user-images.githubusercontent.com/14041622/41086683-5bd826a2-6a6d-11e8-8438-387c2e2de24e.png)

It's so much clearer if you see the function in the middle of integration as a `derivative`.

![fundamental\_theorem\_of\_calculus\_ animation\_](https://user-images.githubusercontent.com/14041622/42437874-ed50bac2-8390-11e8-86f9-280dcd9704d9.gif)

Notice that: In this theorem, the lower boundary `a` is completely "ignored", and the unknown `t` directly changed to `x`.

[►Refer to Khan academy: Fundamental theorem of calculus review](https://www.khanacademy.org/math/ap-calculus-bc/bc-antiderivatives-ftc/modal/a/fundamental-theorem-of-calculus-review) [`►Jump over to have practice at Khan academy: Contextual and analytical applications of integration (calculator active).`](https://www.khanacademy.org/math/ap-calculus-ab/ab-applications-of-integration-new/ab-8-14/e/applications-of-integration-calculator-active)

## `1st FTC & 2nd FTC`

The Fundamental Theorem of Calculus could actually be used in two forms. They have different use for different situations.

(**Notice that boundaries & terms are different**)

![image](https://user-images.githubusercontent.com/14041622/42437797-b8b629a0-8390-11e8-83f3-126e0216f629.png)

## `How to Differentiate Integrals`

We could **CONVERT** the `integral formula` to `Differential formula`, by using the `fundamental theorem of calculus`, and use the Rules we've learnt to solve the differential equations.

[Refer to video from Krista King: PART 2 OF THE FUNDAMENTAL THEOREM OF CALCULUS!](https://www.youtube.com/watch?v=T-J7SkiE39Y\&list=PLJ8OrXpbC-BMObozItpbiZ8f2pjf3qS9M\&index=16)

We got different strategies for different boundaries situation:

* A variable and a number.
* A function and a number.
* Two functions.

▼ Here is formulas for different **boundaries** of integration:

![image](https://user-images.githubusercontent.com/14041622/42652400-e0e5a610-8644-11e8-99d8-4b06106164a3.png)

### Example

![image](https://user-images.githubusercontent.com/14041622/41085306-2eab6b2a-6a69-11e8-9b79-9cf0cc691cc4.png) Solve:

* It's to apply the `boundary situation strategy` of `A variable & a number`: `G'(x) = g(x)`
* Assume the function in the middle of integral is `G'(x) = 3x²+4x`
* Since it's asking for `g'(x)`, so it's differentiate the Integral: `d/dx ʃ G'(x) dt` expression
* So `g'(2) = G'(x) = 3x²+4x = 20`

### Example

![image](https://user-images.githubusercontent.com/14041622/41085593-15943f08-6a6a-11e8-9f71-1789074609b6.png) Solve:

* According to the different `Boundary situation strategies`, here we apply the `A function & a number` strategy: `F'(x) = f[g(x)] · g'(x)`
* So `F'(x) = √(15 - 2x) · (2x)' = 2√(15-2x)`

### Example

![image](https://user-images.githubusercontent.com/14041622/41089874-75a2cefe-6a75-11e8-8ab7-68e7ef3b921d.png) Solve:

* It's asking you to apply the FTC in form of `d/dx ʃ f'(x) dx = f(b) - f(a)`
* So it becomes calculating `F(3) - F(0) = 125 - 1 = 124`

### Example

![image](https://user-images.githubusercontent.com/14041622/42437231-0ab0a2be-838f-11e8-9a34-02def74b102b.png) Solve:

* We could use the `Second Fundamental Theorem of Calculus`:

  ![image](https://user-images.githubusercontent.com/14041622/42437825-d0a50856-8390-11e8-8043-2315c53f2cbd.png)
* which in this case is:

  ![image](https://user-images.githubusercontent.com/14041622/42437405-9a62b2da-838f-11e8-886f-535db3bcc9bb.png)
* And we move the known terms to one side and keep the asking term at another side:

  ![image](https://user-images.githubusercontent.com/14041622/42437475-d27d1138-838f-11e8-99a1-83c730375527.png)