Definite integrals of even and odd functions

A function with $f(-x)=f(x)$ is even, and one with $f(-x)=-f(x)$ is odd. $x^2$, $x^4$ and constants are even; $x$, $x^3$ and $x^5$ are odd.

The graph of an even function is symmetric about the $y$-axis, so the two halves have equal area; the graph of an odd function is symmetric about the origin, so the two halves cancel. For example, $\int_{-1}^{1} x^4\,dx=2\int_{0}^{1} x^4\,dx=\frac{2}{5}$.

Examples

1. 01Which one is an odd function?$\displaystyle$A$\displaystyle 2$B$\displaystyle x^{2}$C$\displaystyle x^{5}$D$\displaystyle x^{2}+1$AnswerHide

   Answer$\displaystyle x^{5}$

   Steps

   1. $\displaystyle (-x)^{5}$$\displaystyle = -x^{5}$
   2. An even power keeps its sign
   3. $\displaystyle (-x)^{2}$$\displaystyle = x^{2}$
2. 02If f(x) is odd, what goes in □?$\displaystyle \int_{-a}^{a} f(x)\,dx = \square$AnswerHide

   Answer$\displaystyle 0$

   Steps

   1. $\displaystyle \int_{-1}^{0} x^{3}\,dx$$\displaystyle = -\frac{1}{4}$$\displaystyle \int_{0}^{1} x^{3}\,dx$$\displaystyle = \frac{1}{4}$
   2. Adding them cancels
   3. $\displaystyle \int_{-1}^{1} x^{3}\,dx$$\displaystyle = 0$
3. 03What is f(-x)?$\displaystyle f(x)=x^{2}$AnswerHide

   Answer$\displaystyle x^{2}$

   Steps

   1. $\displaystyle (-x)^{2}$$\displaystyle = (-1)^{2}x^{2}$$\displaystyle = x^{2}$
4. 04$x^{4}$ is an even function. What is the value of this definite integral?$\displaystyle \int_{-1}^{1} x^{4}\,dx$AnswerHide

   Answer$\displaystyle \frac{2}{5}$

   Steps

   1. $\displaystyle \int_{0}^{1} x^{4}\,dx$$\displaystyle = \frac{1}{5}$
   2. Even, so twice one side
   3. $\displaystyle \int_{-1}^{1} x^{4}\,dx$$\displaystyle = 2 \cdot \frac{1}{5} = \frac{2}{5}$
5. 05What is the value of this definite integral?$\displaystyle \int_{-3}^{3} x^{5}\,dx$AnswerHide

   Answer$\displaystyle 0$

   Steps

   1. $\displaystyle \int_{-3}^{0} x^{5}\,dx$$\displaystyle = -\frac{243}{2}$$\displaystyle \int_{0}^{3} x^{5}\,dx$$\displaystyle = \frac{243}{2}$
   2. Adding them cancels
   3. $\displaystyle \int_{-3}^{3} x^{5}\,dx$$\displaystyle = 0$
6. 06What is f(-x)?$\displaystyle f(x)=x^{5}$AnswerHide

   Answer$\displaystyle -x^{5}$

   Steps

   1. $\displaystyle (-x)^{5}$$\displaystyle = (-1)^{5}x^{5}$$\displaystyle = -x^{5}$

FAQ

Q1What if the function is neither even nor odd, like $x^5+5$?AnswerHide

A

Split it term by term. In $\int_{-2}^{2}(x^5+5)\,dx$, the odd part $x^5$ gives $0$, and the constant $5$ is even, so the result is $2\int_{0}^{2}5\,dx=20$.

Q2Does this work if the interval is not from $-a$ to $a$?AnswerHide

A

No. It holds only on an interval $[-a,\ a]$ that is symmetric about $0$.