If $f$ is an odd function, then $\displaystyle\int_{-a}^{a} f(x)\,dx$ …

Question

If $f$ is an odd function, then $\displaystyle\int_{-a}^{a} f(x)\,dx$ equals

Accepted Answer

C. $0$

1. For an odd function $f(-x)=-f(x)$.
2. By property $\mathbf{P_7}$(ii), the contributions over $[-a,0]$ and $[0,a]$ cancel.
3. Hence $\int_{-a}^{a} f(x)\,dx=0$.
4. Option A is the even-function case.
_Source: NCERT Class 12 Mathematics Ch 7 "Integrals", p.343_

Answer

$2\displaystyle\int_0^a f(x)\,dx$

Answer

$\displaystyle\int_0^a f(x)\,dx$

Answer

$2f(a)$