Answer to Question #238307 in Calculus for moe

Question

True or False

If the function $f$ is odd, then $\displaystyle{\int_{-1}^{1}f(x)dx=0}$

Answer 1

$\text{The solution is } \textit{True} \\ \text{From } \int\limits_{ - 1}^1 {f\left( x \right)} dx = 0. \text{If we let } x = - t \implies dx = - dt \\ \Rightarrow \int\limits_{ - 1}^1 {f\left( x \right)} dx = \int\limits_1^{ - 1} {f\left( { - t} \right)} dt = \int\limits_{ - 1}^1 {f\left( { - x} \right)} dx \\ \text{If } f \text{ is , that is } f\left( { - x} \right) = - f\left( x \right) \\ \Rightarrow \int\limits_{ - 1}^1 {f\left( x \right)} dx + \int\limits_{ - 1}^1 {f\left( { - x} \right)} dx = 0 \\ \text{Which is } \textbf{\textit{True}}$