Calculus How To

Integration of Even and Odd Functions

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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:

  1. 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].
  2. 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:
Integration of Even and Odd Functions

Example question: Find the follow integral:
even definite integral example

definite integral for even function

We can multiply the highlighted area by 2 to get the area of the entire region because the function is symmetrical about the y-axis.

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:
new integral 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:
Integration of Even and Odd Functions Theorem

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.

symmetric about the origin

Graph of an odd function. The two blue regions are identical in area but opposite in sign so cancel each other out,


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

Stephanie Glen. "Integration of Even and Odd Functions" From Calculus for the rest of us!

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