Q. If $f(x) = f(a - x),\ then\int_{0}^{a}{f(x)dx}$ is equal to
A) $\int_{0}^{a}{f(x)dx}$
B) $\frac{a^{2}}{2}\int_{0}^{a}{f(x)dx}$
C) $\frac{a}{2}\int_{0}^{a}{f(x)dx}$
D) $- \frac{a}{2}\int_{0}^{a}{f(x)dx}$
Explanation
(c)
Let $I = \int_{0}^{a}{xf(x)dx}$ …(i)
$\Rightarrow I = \int_{0}^{a}{(a - x)f(a - x)dx}$
$I = \int_{0}^{a}{(a - x)f(x)dx}$
$\lbrack\because f(x) = f(a - x)given\rbrack$
$\Rightarrow I = a\int_{0}^{a}{f(x)dx - \int_{0}^{a}{xf(x)dx\ \ldots(ii)}}$
On adding Eqs.(i) and (ii), we get
$2I = a\int_{0}^{a}{f(x)dx}$
$\Rightarrow I = \frac{a}{2}\int_{0}^{a}{f(x)dx}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.