Integrating Even and Odd Functions

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 $f(\text{−}x)=f(x)$ for all $x$ in the domain—that is, the graph of the curve is unchanged when $x$ is replaced with −$x$. The graphs of even functions are symmetric about the $y$-axis. An odd function is one in which $f(\text{−}x)=\text{−}f(x)$ for all $x$ 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 −$a$ to $a$, involve two equal areas, because they are symmetric about the $y$-axis. Integrals of odd functions, when the limits of integration are similarly $\left[\text{−}a,a\right],$ evaluate to zero because the areas above and below the $x$-axis are equal.

Integrals of Even and Odd Functions

For continuous even functions such that $f(\text{−}x)=f(x),$

${\displaystyle\int }_{\text{−}a}^{a}f(x)dx=2{\displaystyle\int }_{0}^{a}f(x)dx.$

For continuous odd functions such that $f(\text{−}x)=\text{−}f(x),$

${\displaystyle\int }_{\text{−}a}^{a}f(x)dx=0.$
It may be useful to recall how to quickly determine whether a function is even, odd or neither.

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

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

1. $f(x)=-5x^4+7x^2-2$
2. $f(x)=2x^5-4x+5$
3. $f(x)=\large{\frac{3x}{x^2+1}}$

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

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

Example: Integrating an Even Function

Integrate the even function ${\displaystyle\int }_{-2}^{2}(3{x}^{8}-2)dx$ 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 $-5 \sin x$ over the interval $\left[\text{−}\pi ,\pi \right].$

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

Try It

Integrate the function ${\displaystyle\int }_{-2}^{2}{x}^{4}dx.$