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 .
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.
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.
Larson, R. & Edwards, B. (2009). Calculus (9th Edition). Cengage Learning.
Stephanie Glen. "Integration of Even and Odd Functions" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/integration-of-even-and-odd-functions/
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