Integrating Even and Odd Functions: Methods

Definite integral of an odd function (KristaKingMath)
Definite integral of an odd function (KristaKingMath)

StudySmarter – The all-in-one study app.

4.8 • +11k Ratings

More than 3 Million Downloads

Free

Americas

Europe

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 OriginalsNotice 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 OriginalsA graph with a reflection across…

Explore our app and discover over 50 million learning materials for free.

Save the explanation now and read when you’ve got time to spare.

Save

Lerne mit deinen Freunden und bleibe auf dem richtigen Kurs mit deinen persönlichen Lernstatistiken

Jetzt kostenlos anmelden

Nie wieder prokastinieren mit unseren Lernerinnerungen.

Jetzt kostenlos anmelden

Functions can have some fascinating graphs—hidden patterns, fantastic shapes, and symmetries. Take, for instance, the graph below.

Notice something? It looks like we put a mirror on the y-axis! Let’s now take a look at another graph:

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!

Functions that have these types of symmetries receive unique names. Furthermore, these symmetries can be used in our favor when integrating these 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!

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.

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:

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.

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:

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!

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.

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.

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!

Even functions are functions whose output does not change if you change the sign of their input.

Odd functions are functions such that only the sign of their output changes if you change the sign of their input.

To prove the formula for integrating even and odd functions, you use the basic properties of integrals along with the properties of even and odd functions.

First look at the interval of integration in order to use the formula for integrating odd functions. If the interval is of the form [-a,a] then the integral of the odd function is equal to zero.

First look at the interval of integration in order to use the formula for integrating even functions. If the interval is of the form [-a,a] then the integral of the even function is equal to twice the integral over the interval [0,a].

Question

Even functions are also known as:

Show answer

Answer

Symmetric functions.

Show question

Question

Odd functions are also known as:

Show answer

Answer

Antisymmetric functions.

Show question

Question

Let \(f\) be an even function that is integrable over the interval \([-a,a]\). Which of the following formulas holds true?

Show answer

Answer

\(\int_{-a}^a f(x) \, \mathrm{d}x=2\int_0^a f(x)\, \mathrm{d}x.\)

Show question

Question

Find the value of \(\int_{-1}^1 10x^{4}+6x^{2}+1 \, \mathrm{d}x\).

Show answer

Answer

\(10\).

Show question

Question

Find the value of \(\int_{-3}^3 (x^7-5x^5+3x^3+x) \, \mathrm{d}x\).

Show answer

Answer

\(0\).

Show question

Question

Find the value of \(\int_{-\pi}^{\pi} \sin x \, \mathrm{d}x\).

Show answer

Answer

\(0\).

Show question

Question

Find the value of \( \int_{\pi/2}^{-\pi/2} \cos x \, \mathrm{d}x\).

Show answer

Answer

\(-2\).

Show question

of the users don’t pass the Integrating Even and Odd Functions quiz! Will you pass the quiz?

Start Quiz

How would you like to learn this content?

94% of StudySmarter users achieve better grades.

Sign up for free!

94% of StudySmarter users achieve better grades.

Sign up for free!

How would you like to learn this content?

Free math cheat sheet!

Everything you need to know on . A perfect summary so you can easily remember everything.

Be perfectly prepared on time with an individual plan.

Test your knowledge with gamified quizzes.

Create and find flashcards in record time.

Create beautiful notes faster than ever before.

Have all your study materials in one place.

Upload unlimited documents and save them online.

Identify your study strength and weaknesses.

Set individual study goals and earn points reaching them.

Stop procrastinating with our study reminders.

Earn points, unlock badges and level up while studying.

Create flashcards in notes completely automatically.

Create the most beautiful study materials using our templates.

Sign up to highlight and take notes. It’s 100% free.

Save explanations to your personalised space and access them anytime, anywhere!

Sign up with Email Sign up with Apple

By signing up, you agree to the Terms and Conditions and the Privacy Policy of StudySmarter.

Already have an account? Log in

You are watching: Integrating Even and Odd Functions: Methods. Info created by THVinhTuy selection and synthesis along with other related topics.

Rate this post

Related Posts