https://youtu.be/NbwJOQnT1eM

In this video I go over the theorem on integrating symmetric functions which greatly simplifies integration. For even functions the integral from -a to a is just two times the integral from 0 to a. For odd functions, the integral from -a to a is simply zero. In this video I also provide a simple proof of this theorem while utilizing the substitution rule for integrals and properties of definite integrals.

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# Symmetry in Integrals

![Symmetry in Integrals Examples.jpeg](https://files.peakd.com/file/peakd-hive/mes/re0wUiY4-Symmetry20in20Integrals20Examples.jpeg)

## Integrals of Symmetric Functions Theorem

Suppose f is continuous on [-a, a].

(a) If f is even, meaning f(-x) = f(x), then:

![image.png](https://files.peakd.com/file/peakd-hive/mes/OqcxkUx9-image.png)

(b) If f is odd, meaning f(-x) = - f(x), then:

![image.png](https://files.peakd.com/file/peakd-hive/mes/iZbdoKM2-image.png)

**Proof:**

![image.png](https://files.peakd.com/file/peakd-hive/mes/eDfFz48l-image.png)

![image.png](https://files.peakd.com/file/peakd-hive/mes/VTUdvmOZ-image.png)

![image.png](https://files.peakd.com/file/peakd-hive/mes/tGewgocT-image.png)

![image.png](https://files.peakd.com/file/peakd-hive/mes/EHQutdKn-image.png)