# Antiderivatives

> That's easier being said, **THE ANTIDERIVATIVES IS THE INDEFINITE INTEGRALS.**

Let's make it more intuitive (might not be accurate but good for learning):

* `Antiderivatives` is the **RESULT**.
* `Indefinite integrals` is the **OPERATION**.

(They're saying the same thing)

And, just for refreshing:

* `Anti-derivatives`: Means the **Original function** where the derivative is from.
*   `Indefinite integrals`: **Indefinite**  means `not-defined`,  means both **BOUNDARIES** are not defined. That's why the **symbol** is without any number but `ʃ` alone.

    ![image](https://user-images.githubusercontent.com/14041622/41083178-1b730172-6a63-11e8-8dea-f0e3d6f8a78b.png)

## Why is the Indefinite Integral so confusing

It's a simple reason: **Because they use the** `f(x)` **in the Integral expression, but actually it means** `f'(x)`**!**

You all know the expression of indefinite integral is `ʃ f(x) dx`, But actually it should be `ʃ f'(x) dx`, which means the function appears in the middle is a derivative, from somewhere. And the **mission** of that integral, is to find the `f(x)` the original function of the derivative!

So trust me, the world would be much nicer if you always see it as the expression as below:

![image](https://user-images.githubusercontent.com/14041622/41083446-cf7a6b6a-6a63-11e8-9b58-c6141679b285.png)

## Why is Antiderivative so confusing too

Because your first impression of the antiderivative is that `is it anti- something?` `Anti-` is a reverse, `derivative` is also a transform of something, so putting them together is really a horrible idea because it seems leading to nowhere.

Now here is the mojo, things would be much nicer if you see and call an `antiderivative` as: **The original function** `f(x)`

It may not be accurate, but good enough to proceed to next stage of study.

## Review of Antiderivatives

> Before you proceed to the next, you really want figure out completely what an `antiderivative` means with respect to the `Integration`.

[!! Refer to video from The Organic Chemistry Tutor: Antiderivatives](https://www.youtube.com/watch?v=Smp1WJjfUvM\&list=PL0o_zxa4K1BWYThyV4T2Allw6zY0jEumv\&t=0s\&index=108)

Here are a few examples to quick review what is antiderivative:

![image](https://user-images.githubusercontent.com/14041622/40964276-45f36c52-68dd-11e8-8e46-cc8fc947b5ed.png)

## How to understand this reverse process

Doing an `Integration`, is actually to find the `antiderivative`.

At the example below, you will find it makes so much sense if you **FIX YOUR EYES** only onto the **MIDDLE** part of the integration formula, the part between `ʃ` & `d/dx`.

![image](https://user-images.githubusercontent.com/14041622/40964653-3d241602-68de-11e8-8f3a-6e4a5963b363.png)

### Example

![image](https://user-images.githubusercontent.com/14041622/41082572-64fe8656-6a61-11e8-94ef-9f77ac4147b0.png) Solve:

* First need to clear your mind, in this case:
  * The antiderivative is the "original function" `f(x)` we have learned.
  * The function `f(x)` here is actually the derivative, which is `f'(x)` we've learned.
* Just think it as matching function and its derivative, everything will be tackled real quick.

### Example

![image](https://user-images.githubusercontent.com/14041622/41082951-862bab00-6a62-11e8-9264-820e22eda246.png) Solve:

* First need to make sense of the terms:
  * The function appears in the `Integral` expression, is actually the `derivative`
  * The "result" is just the `Original function` where the derivative comes from, or you could call the result is the antiderivative.
* Just think it as matching function and its derivative, everything will be tackled real quick.
