If f(a + b - x) = f(x), then _a^b x ,f(x) ,dx = - Vidyadip

MCQ

If $f(a + b - x) = f(x),$ then $\int_a^b {x\,f(x)\,dx = } $

A
$\frac{{a + b}}{2}\int_a^b {f(b - x)\,dx} $
✓
$\frac{{a + b}}{2}\int_a^b {f(x)\,dx} $
C
$\frac{{b - a}}{2}\int_a^b {f(x)\,dx} $
D
None of these

✓

Answer

Correct option: B.

$\frac{{a + b}}{2}\int_a^b {f(x)\,dx} $

b
(b) Since $I = \int_a^b {xf(x)dx = \int_a^b {(a + b - x)f(a + b - x)dx} } $

==> $I = \int_a^b {(a + b)} f(x)dx - \int_a^b {xf(x)dx} $

$\left\{ \because f(a+b-x)=f(x)\text{ given} \right\}$

==> $2I = (a + b)\int_a^b {f(x)dx} $

==> $I = \int_a^b {x\,f(x)dx = \frac{{a + b}}{2}\int_a^b {f(x)dx} } $.

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