Functions can have some fascinating graphs—hidden patterns, fantastic shapes, and symmetries. Take, for instance, the graph below.
Graph of a symmetric quartic function - StudySmarter Originals
Notice something? It looks like we put a mirror on the y-axis! Let's now take a look at another graph:
Graph of a symmetric pattern using the cosine function - StudySmarter Originals
A graph with a reflection across the x-axis is not a function because it fails the vertical line test. However, if we slide the portion below the x-axis to the left, we now have rotational symmetry about the origin!
Graph of the cosine pattern made into a function - StudySmarter Originals
Functions that have these types of symmetries receive unique names. Furthermore, these symmetries can be used in our favor when integrating these 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.
A function is said to be an even function, or symmetric function, if for all x-values of its domain.
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?
A function is said to be an odd function, or antisymmetric function, if for all x-values of its domain.
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.
Keep in mind that some functions are neither even nor odd!
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 be a function that is integrable in the interval . If is an even function, then the following formula holds true:
Let's look at the area between an even function and the positive x-axis.
The area between an even function and the positive x-axis - StudySmarter Originals
We can compare it to the area between the same function and the negative x-axis.
The area between the same even function and the negative x-axis - StudySmarter Originals
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 be a function that is integrable in the interval . If is an odd function, then the following formula holds true:
Let's take a look at the definite integral of an odd function.
Integral of an odd function - StudySmarter Originals
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.
Find the value of the definite integral:
Solution:
We begin by inspecting if the given function is even or odd.
Set equal to the integrand of the definite integral.
Evaluate and simplify using the properties of exponents.
We have verified that the function is even because . We can now use the formula for integrating even functions.
Integrate using Basic Integration Formulas.
Evaluate and simplify.
Using this formula is that the evaluation when is relatively simple. Let's see another example.
Find the value of the definite integral: .
Solution:
Once again, we begin by inspecting if the function is even or odd.
Set equal to the integrand of the definite integral.
Evaluate and simplify using the properties of exponents.
Factor out -1.
We have verified the function is odd because . We can now use the formula for integrating odd functions.
Note that we did not have to use any other integration rule!
Here is a quick tip for Polynomial Functions! If all exponents of the Polynomial Function are even numbers (or constant terms), then the function is even. Likewise, if all exponents of the polynomial function are odd, then the function is odd. Using this shortcut, you don't have to evaluate f(-x) and you can quickly find out which formula to use!
Integrating Even and Odd Functions - Key takeaways
- We can classify some functions into even functions or odd functions.
- Even functions are functions such that for every x in its domain.
- Odd functions are functions such that for every x in its domain.
- If is an even function that is integrable on the interval then .
- If is an odd function that is integrable on the interval then .
- Even functions are also known as symmetric functions. Odd functions are also known as antisymmetric functions.
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