$\displaystyle\int_0^{2a} f(x)\,dx$ equals $2\displaystyle\int_0^{a} f…

Question

$\displaystyle\int_0^{2a} f(x)\,dx$ equals $2\displaystyle\int_0^{a} f(x)\,dx$ precisely when

Accepted Answer

B. $f(2a-x)=f(x)$

1. Property $\mathbf{P_6}$ splits $\int_0^{2a}$ into $\int_0^a f(x)+\int_0^a f(2a-x)$.
2. If $f(2a-x)=f(x)$, the two integrals are equal, giving $2\int_0^a f(x)\,dx$.
3. If instead $f(2a-x)=-f(x)$, the result is $0$ (option A's case).
_Source: syllabus Mathematics Ch 7 "Integrals", p.342_

Answer

$f(2a-x)=-f(x)$

Answer

$f$ is an odd function

Answer

$f(a-x)=f(x)$

$\displaystyle\int_0^{2a} f(x)\,dx$ equals $2\displaystyle\int_0^{a} f(x)\,dx$ precisely when

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