### Learning Outcomes

- Apply the integrals of odd and even functions

We saw in Module 1: Functions and Graphs that an even function is a function in which [latex]f(\text{−}x)=f(x)[/latex] for all [latex]x[/latex] in the domain—that is, the graph of the curve is unchanged when [latex]x[/latex] is replaced with −[latex]x[/latex]. The graphs of even functions are symmetric about the [latex]y[/latex]-axis. An odd function is one in which [latex]f(\text{−}x)=\text{−}f(x)[/latex] for all [latex]x[/latex] in the domain, and the graph of the function is symmetric about the origin.

Integrals of even functions, when the limits of integration are from −[latex]a[/latex] to [latex]a[/latex], involve two equal areas, because they are symmetric about the [latex]y[/latex]-axis. Integrals of odd functions, when the limits of integration are similarly [latex]\left[\text{−}a,a\right],[/latex] evaluate to zero because the areas above and below the [latex]x[/latex]-axis are equal.

### Integrals of Even and Odd Functions

For continuous even functions such that [latex]f(\text{−}x)=f(x),[/latex]

For continuous odd functions such that [latex]f(\text{−}x)=\text{−}f(x),[/latex]

### Recall: How to determine whether a function is even, odd or neither

Determine whether each of the following functions is even, odd, or neither.

- [latex]f(x)=-5x^4+7x^2-2[/latex]
- [latex]f(x)=2x^5-4x+5[/latex]
- [latex]f(x)=\large{\frac{3x}{x^2+1}}[/latex]

To determine whether a function is even or odd, we evaluate [latex]f(−x)[/latex] and compare it to [latex]f(x)[/latex] and [latex]−f(x)[/latex].

- [latex]f(−x)=-5(−x)^4+7(−x)^2-2=-5x^4+7x^2-2=f(x)[/latex]. Therefore, [latex]f[/latex] is even.
- [latex]f(−x)=2(−x)^5-4(−x)+5=-2x^5+4x+5[/latex]. Now, [latex]f(−x)\ne f(x)[/latex]. Furthermore, noting that [latex]−f(x)=-2x^5+4x-5[/latex], we see that [latex]f(−x)\ne −f(x)[/latex]. Therefore, [latex]f[/latex] is neither even nor odd.
- [latex]f(−x)=\frac{3(−x)}{((−x)^2+1)}=\frac{-3x}{(x^2+1)}=−\left[\frac{3x}{(x^2+1)}\right]=−f(x)[/latex]. Therefore, [latex]f[/latex] is odd.

### Example: Integrating an Even Function

Integrate the even function [latex]{\displaystyle\int }_{-2}^{2}(3{x}^{8}-2)dx[/latex] and verify that the integration formula for even functions holds.

Watch the following video to see the worked solution to Example: Integrating an Even Function.

### Example: Integrating an Odd Function

Evaluate the definite integral of the odd function [latex]-5 \sin x[/latex] over the interval [latex]\left[\text{−}\pi ,\pi \right].[/latex]

Watch the following video to see the worked solution to Example: Integrating an Odd Function.

### Try It

Integrate the function [latex]{\displaystyle\int }_{-2}^{2}{x}^{4}dx.[/latex]