The theorem of “Integration of Even and Odd Functions” is a way to find integrals for odd and even functions. It’s **a method that makes some challenging definite integrals easier to find. **In order to use it, you have to be evaluating a function over an interval that’s either symmetric about the origin or symmetric about the y-axis [1].

In order to use this theorem to simplify finding a definite integral:

**Your closed interval must be in the form [-a, a].**In other words, your interval must have the same number “a” like [-3, 3] or [-π, π].**You must have an even or odd function.**If you aren’t sure use the Test for Even and Odd Functions to check before continuing.

## Integration of Even and Odd Functions: Even Example

**1. Even functions Rule: **

**Example question:** Find the follow integral:

Step 1:

**Rewrite the integral to cover the positive half of the region**(shown in the above image). For this example, instead of integrating from -1 to 1, we’re integrating from 0 to 1:

Step 2: **Solve the integral** (I used Symbolab’s calculator):

Solving the integral, we get 8/7 ≈ 1.14285.

Step 3: **Multiply the solution from Step 2 by 2:**

8/7 * 2 = 16/7 ≈ 2.28571.

*That’s it!*

## 2. Odd functions Definite Integral Example

This is the *easiest definite integral you’ll ever calculate* (actually, you don’t even have to do any calculations…read on!).

The formula is:

This part of the theorem is easier than for even functions. It simply states that **if your function is symmetric about the origin for interval [-a, a], the definite integral is going to be zero.**

## References

Larson, R. & Edwards, B. (2009). Calculus (9th Edition). Cengage Learning.

**CITE THIS AS:**

**Stephanie Glen**. "Integration of Even and Odd Functions" From

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