$\displaystyle\int_0^{2a} f(x)\,dx$ equals $2\displaystyle\int_0^{a} f(x)\,dx$ precisely when
A$f(2a-x)=-f(x)$
B$f(2a-x)=f(x)$
C$f$ is an odd function
D$f(a-x)=f(x)$
Answer & Solution
Correct 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: NCERT Class 12 Mathematics Ch 7 "Integrals", p.342_
Related questions
$\displaystyle\int_2^3 \dfrac{x\,dx}{x^2+1}$ equalsIf $f(a+b-x)=f(x)$, then $\displaystyle\int_a^b x\,f(x)\,dx$ is equal to$\displaystyle\int \frac{dx}{e^{x}+e^{-x}}$ is equal toA rational function $\dfrac{P(x)}{Q(x)}$ is called proper when$\displaystyle\int_0^{\pi/4} \tan x\,dx$ equals$\displaystyle\int_0^{1} x e^{x^2}\,dx$ equalsThe value of $\displaystyle\int_0^{\pi/2} \log\!\left(\dfrac{4+3 in x}{4+3\cos x}\right)dxUsing property $\mathbf{P_2}$, $\displaystyle\int_{-1}^{2} |x^3-x|\,dx$ equals