> 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-395949004.md).

# Partial fractions → Log Rule

> A technique for integrating `Rational functions`.

[`▶ Jump back to previous note on Partial fractions.`](https://github.com/solomonxie/solomonxie.github.io/issues/44#issuecomment-404754134)

[▶Refer to Khan academy: Partial fraction expansion to evaluate integral](https://www.khanacademy.org/math/ap-calculus-bc/bc-antiderivatives-ftc/modal/v/partial-fraction-expansion-to-integrate)

## Example

![image](https://user-images.githubusercontent.com/14041622/41189233-0e9b8bfe-6bfd-11e8-94f0-88bae4d86f14.png) This process is to break down the `Rational Function` to some simple fractions, which assume there are `A & B` leads to a system of equation:

* `(A+B)·x + (B-A) = 1·x + (-4)`
* So `(A+B) = 1` and `(B-A) = -4`, which gets us `A = 5/2` & `B = -3/2`

Strategy:

* Look at the `Nominator` & `Dominator`'s degrees.
* If the dominator's degrees are higher or equal than the nominator, we do `Long division of polynomial` to downgrade it.
* Try to `factorize` the `dominator` if you can.
  * Assume two variables `A & B`
  * Apply the `Partial Fraction Expansion` technique.
* Apply the basic `Log Rule` to solve the parts.

## Example

![image](https://user-images.githubusercontent.com/14041622/41226555-69edc06e-6da4-11e8-88e7-59a1c0cacf78.png) Solve: ![image](https://user-images.githubusercontent.com/14041622/41226940-a442378a-6da5-11e8-9d67-66a8dc735ecd.png)

## Example

![image](https://user-images.githubusercontent.com/14041622/41226173-55f8981e-6da3-11e8-8acc-626aef8cef42.png) Solve: [Refer to Symbolab.](https://www.symbolab.com/solver/step-by-step//int%20/frac%7Bx-1%7D%7B2x%2B4%7Ddx) ![image](https://user-images.githubusercontent.com/14041622/41226547-5ee07d6a-6da4-11e8-906f-6f1c80876de9.png)