Question

If $f(a + b - x) = f(x),$ then $\int^b_a \, xf (x) dx$ is equal to:

• A. $\frac{a +b}{2} \int^b_a f(b - x)dx$
• B. $\frac{a +b}{2} \int^b_a f( x)dx$
• C. $\frac{b - a }{2} \int^b_a f( x)dx$
• D. none of these

Answer 1

Answer: (B)

Given f(a + b - x) = f(x)
$\Rightarrow \, a +b - x = x \, \Rightarrow \, x = \frac{a + b}{2}$
Let $I = \int^b_a xf (x) dx = \frac{a+ b}{2} \int^b_a f(x) dx $