f(x) = f(2 - x), then _ ,0.5 ^ ,1.5 ,xf(x) ,dx equals - Vidyadip

MCQ

$f(x) = f(2 - x),$ then $\int_{\,0.5}^{\,1.5} {\,xf(x)\,dx} $ equals

A
$\int_{\,0}^{\,1} {\,f(x)\,dx} $
✓
$\int_{\,0.5}^{\,1.5} {\,f(x)\,dx} $
C
$2\int_{\,0.5}^{\,1.5} {\,f(x)\,dx} $
D
$0$

✓

Answer

Correct option: B.

$\int_{\,0.5}^{\,1.5} {\,f(x)\,dx} $

b
(b) $I = \int_{0.5}^{1.5} {xf\,(x)\,dx = \int_{0.5}^{1.5} {(2 - x)f(2 - x)\,dx} } $,

$\left[ \because \int_{a}^{b}{f(x)dx=\int_{a}^{b}{f(a+b-x)dx}} \right]$

$ = \int_{0.5}^{1.5} {(2 - x)f(x)\,dx} = 2\int_{0.5}^{1.5} {f(x)\,dx - I} $

==>$I = \int_{0.5}^{1.5} {f(x)\,dx} $.

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