Q. If $f(a + b - x) = f(x),\ then\int_{a}^{b}xf(x)dx$ is equal to
A) $\frac{a + b}{2}\int_{a}^{b}{f(b - x)dx}$
B) $\frac{a + b}{2}\int_{a}^{b}{f(x)dx}$
C) $\frac{b - a}{2}\int_{a}^{b}{f(x)dx}$
D) $\frac{a + b}{2}\int_{a}^{b}{f(a + b + x)dx}$
Explanation
(b)
Let $I = \int_{a}^{b}{x\ f(x)dx\ }\ \ \ \$… (i)
$\Rightarrow \ I = \int_{a}^{b}{(a + b - x)f(x)dx}$
[since, $f(a + b - x) = f(x),$given]
$\Rightarrow \ I = (a + b)\int_{a}^{b}{f(x) - I\ \lbrack from\ Eq.(i)\rbrack}$
$\Rightarrow \ 2I = (a + b)\int_{a}^{b}{f(x)dx}$
$\Rightarrow \ I = \left( \frac{a + b}{2} \right)\int_{a}^{b}{f(x)dx}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.