# Improper Integral

After learning `Definite Integral`, `Indefinite Integral`, now it's `Improper Integral`. The major difference between them is their `Boundaries`.

> The `improper integral` means the integral's boundary or boundaries are infinite, **∞** (or -∞).

[Refer to Khan academy: Introduction to improper integrals](https://www.khanacademy.org/math/ap-calculus-bc/bc-antiderivatives-ftc/modal/v/introduction-to-improper-integrals) [Refer to Improper Integrals (KristaKingMath)](https://www.youtube.com/watch?v=DUowajlKIKA)

It looks so fearful yet not too hard to understand.

![image](https://user-images.githubusercontent.com/14041622/41109490-b6794e24-6aa9-11e8-8835-9ba1a704bfc6.png)

## `Types of Improper Integral`

[Refer to video from ProfRobBob: Improper Integrals 5 Examples](https://www.youtube.com/watch?v=dzOxbMnaZLo)

There're 6 cases of different improper integral:

*   Case 1: From a `constant` to `positive infinity`.

    ![image](https://user-images.githubusercontent.com/14041622/41142396-2457bd62-6b28-11e8-80c6-1f93eaed3c6e.png)
*   Case 2: From `negative infinity` to a `constant`.

    ![image](https://user-images.githubusercontent.com/14041622/41142107-161669ac-6b27-11e8-8b6f-d45fb848351f.png)
*   Case 3: From `negative infinity` to `positive infinity`.

    ![image](https://user-images.githubusercontent.com/14041622/41142432-3f663dd6-6b28-11e8-8ab0-10ce0f0474cb.png)
*   Case 4: From `0` to `e`.

    ![image](https://user-images.githubusercontent.com/14041622/41142494-75b483d4-6b28-11e8-9985-9950ad73efe8.png)
*   Case 5: From a `constant` to a `constant`, but has an `infinite discontinuity`.

    ![image](https://user-images.githubusercontent.com/14041622/41142565-b8ca0784-6b28-11e8-97a8-5040f5946000.png)
* Case 6:&#x20;

## `Convergent` & `Divergent`

We can call an `improper integral`:

* `Divergent`: When the **limit** of the improper integral **DOES NOT EXIST**.
* `Convergent`: When the **limit** of the improper **integral EXISTS**.

## `Solve Improper Integrals`

Basic Strategy:

* Replace the `infinite` as a variable, etc. `t`
* Rewrite the expression as taking the limit of the Integral, whereas the `t → ∞`
* Calculate the Integral with a normal variable first, and gets the result function.
* Calculate the limit of the function

### Type 1

![image](https://user-images.githubusercontent.com/14041622/41143766-1e3a65ce-6b2d-11e8-831b-b4b0a34fb260.png) Solve: ![image](https://user-images.githubusercontent.com/14041622/41143689-cf7ded34-6b2c-11e8-89d1-149ed496ac65.png)

### Type 2

![image](https://user-images.githubusercontent.com/14041622/41144173-95f24554-6b2e-11e8-9fc7-9b541e058192.png) Solve:

*   Rewrite the improper integral to limit form:

    ![image](https://user-images.githubusercontent.com/14041622/41144216-bd090d76-6b2e-11e8-9518-4c0cfd1c0ccd.png)
* Do the basic calculation.
*   The key point is:

    ![image](https://user-images.githubusercontent.com/14041622/41144261-ecb0c5e6-6b2e-11e8-967d-924b37cf87da.png)

### Type 5

![image](https://user-images.githubusercontent.com/14041622/41143759-15bde20e-6b2d-11e8-84a4-c0d5f18a6749.png) Solve: ![image](https://user-images.githubusercontent.com/14041622/41143923-ae356d40-6b2d-11e8-9dac-b3c5554a4e40.png)
