# Integration using Trig identities

> The goal is to convert the **multiplication** of terms to be **addition** of simple terms.

[`▶ Cheatsheet on previous note: Basic Integral Rules`](https://github.com/solomonxie/solomonxie.github.io/issues/49#issuecomment-395356656) [`▶ Cheatsheet on previous note:Basic Differential Rules`](https://github.com/solomonxie/solomonxie.github.io/issues/49#issuecomment-390102382) [`▶ Cheatsheet on previous note: All trig identities`](https://github.com/solomonxie/solomonxie.github.io/issues/44#issuecomment-377684464) [`▶ Back to previous note on: U-substitution → Chain Rule`](https://github.com/solomonxie/solomonxie.github.io/issues/49#issuecomment-395677669)

[Refer to Khan academy: Integrating using trigonometric identities](https://www.khanacademy.org/math/integral-calculus/ic-integration#ic-integration-with-trig-identities)

## Example

![image](https://user-images.githubusercontent.com/14041622/46219793-d9a9a400-c37a-11e8-8075-f92004958982.png) Solve:

*   We can identify there's a `Integral trig rule` we can apply on this problem:

    ![image](https://user-images.githubusercontent.com/14041622/46241862-a4409d00-c3f2-11e8-9618-c1e1e7af7186.png)
*   Therefore we're gonna organize it a bit with `u-substitution` and get the answer:

    ![image](https://user-images.githubusercontent.com/14041622/46241876-e964cf00-c3f2-11e8-9b81-8a362400cc60.png)

## Example

![image](https://user-images.githubusercontent.com/14041622/46242010-4eb9bf80-c3f5-11e8-83c3-3bd77a63566d.png) Solve:

*   This integrand has an **odd** power of `sin(x)`, so we're gonna leave `sin(x)` alone and make an  **even** power for `sin(x)`, then make `u-substitution` upon it:

    ![image](https://user-images.githubusercontent.com/14041622/46242032-aeb06600-c3f5-11e8-8ee1-382141b49e05.png)
*   And also we need to change the boundaries for the integral:

    ![image](https://user-images.githubusercontent.com/14041622/46242042-ef0fe400-c3f5-11e8-9f53-ffaafeac704a.png)
*   Now we can start to integrate:

    ![image](https://user-images.githubusercontent.com/14041622/46242061-25e5fa00-c3f6-11e8-8aed-148deb16aa6b.png)

## Example

![image](https://user-images.githubusercontent.com/14041622/46242288-92162d00-c3f9-11e8-9088-651c530a46ee.png) Solve:

*   We're gonna use a `trig-integral-rule` and a `trig identity`:

    ![image](https://user-images.githubusercontent.com/14041622/46242353-a3136e00-c3fa-11e8-8c83-8152e3e32687.png)
*   Then apply them:

    ![image](https://user-images.githubusercontent.com/14041622/46242364-ca6a3b00-c3fa-11e8-8646-caa17e3852f6.png)
