Integrating Even and Odd Functions

What are even and odd functions?

We can classify some functions into even functions or odd functions. But what does this mean? Let's take a look at each definition.

In other words, an even function is a function whose output does not change if we change the sign of its input. What about odd functions?

In contrast, if we change the sign of the input of an odd function, we will get the opposite of its output. We can take advantage of this symmetry when finding definite integrals of even or odd functions.

Definite integrals of even functions

For graphs of even functions, every value to the left of the y-ais mirrors the value to the right of it. This characteristic gives us the formula for definite integrals of even functions.

Let's look at the area between an even function and the positive x-axis.

We can compare it to the area between the same function and the negative x-axis.

Note how the areas are the same; they are just reflected over the y-axis. This observation means that we can find the area of the whole interval by multiplying either of these areas by 2. Typically, we use the area above the positive x-axis, which gives us the formula for integrating even functions:

Definite integrals of odd functions

The graph of an odd function is like an even function, but the mirror values are negative. Below is the formula for integrating odd functions.

Let's take a look at the definite integral of an odd function.

Note how the areas are the same again, but now they are reflected over both axes. In this case, one area is the negative of the other. Therefore, if we add them together, the result is equal to 0. From this fact, we obtain the formula for definite integrals of odd functions:

Proofs of the integrals of even and odd functions

We can use the properties of even and odd functions to prove the formulas for integrating even and odd functions. Let's jump into it!

Proof of the integral of an even function

Let be an even function. Consider the definite integral .

We can split this integral into two intervals using the properties of integrals.

Since is an even function, we can replace with in the first term of the right-hand side of the equation.

We now do an u-substitution (see Integration by Substitution) in the same term by letting . This way and the lower limit of integration becomes .

We now use the minus sign inside the integral to flip the limits of the definite integral.

Since the integrals involved in the above equation are all definite integrals, the variable of integration does not matter at all. Hence, the first and the second term of the right-hand side of the equation are equal.

This equivalence gives us the formula for integrating even functions.

Proof of the integral of an odd function

Let be an odd function. Consider the definite integral .

We will split this integral as well.

Since is an odd function, we can replace with . This substitution is essentially changing the sign of the function from positive to negative and back to positive again.

We can now proceed in the same way as we did for the integral of an even function by doing the same u-substitution and simplifying.

The two terms on the right-hand side of the equation are the negative of each other. So their sum is equal to 0. This simplification gives us the formula for integrating odd functions.

Examples of even and odd function integration

When using the formulas for integrating even and odd functions, we need to be sure that our function is even or odd. Let's see how this is done.

Using this formula is that the evaluation when is relatively simple. Let's see another example.

Note that we did not have to use any other integration rule!