Question

If $f\left(x\right)$ is an odd function, then ${\int }_{-a}^{a}f\left(x\right)dx$ is

• A. $2{\int }_0^{a}f\left(x\right)dx$
• B. ${\int }_0^{a}f\left(x\right)dx$
• C. $0$
• D. ${\int }_0^{a}f(a-x)dx$

Answer 1

The correct option is C $0$

${\int }_{-a}^{a}f\left(x\right)dx$
$={\int }_{-a}^{0}f\left(x\right).dx+{\int }_0^{a}f\left(x\right).dx$
$=F\left(0\right)-F(-a)+F\left(a\right)-F\left(0\right)$
$=F\left(a\right)-F(-a)$
Since $f\left(x\right)$ is an odd function, its integral will be an even function.
Therefore $F\left(x\right)=F(-x)$
Hence $F\left(a\right)=F(-a)$ or $F\left(a\right)-F(-a)=0$ or $I=0$