If $f(a+b-x)=f(x)$, then $\displaystyle\int_a^b x\,f(x)\,dx$ is equal to
A$\dfrac{a+b}{2}\displaystyle\int_a^b f(b-x)\,dx$
B$\dfrac{a+b}{2}\displaystyle\int_a^b f(b+x)\,dx$
C$\dfrac{b-a}{2}\displaystyle\int_a^b f(x)\,dx$
D$\dfrac{a+b}{2}\displaystyle\int_a^b f(x)\,dx$
Answer & Solution
Correct answer: D. $\dfrac{a+b}{2}\displaystyle\int_a^b f(x)\,dx$
1. Let $I=\int_a^b x f(x)\,dx$. By $\mathbf{P_3}$, $I=\int_a^b (a+b-x)f(a+b-x)\,dx$.
2. Given $f(a+b-x)=f(x)$, this is $\int_a^b (a+b-x)f(x)\,dx$.
3. So $2I=(a+b)\int_a^b f(x)\,dx$.
4. Hence $I=\dfrac{a+b}{2}\int_a^b f(x)\,dx$.
_Source: NCERT Class 12 Mathematics Ch 7 "Integrals", p.351_
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