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} $
- DNone of these
==> $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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