If f(a+b-x)=f(x), then _a^b x f(x) d x is equal to - Vidyadip

MCQ

If $f(a+b-x)=f(x)$, then $\int_a^b x f(x) d x$ is equal to

A
$\frac{a+b}{2} \int_a^b f(b-x) d x$
B
$\frac{a+b}{2} \int_a^b f(b+x) d x$
C
$\frac{b-a}{2} \int_a^b f(x) d x$
✓
$\frac{a+b}{2} \int_a^b f(x) d x$

✓

Answer

Correct option: D.

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

(D) $\frac{a+b}{2} \int_a^b f(x) d x$
Let, $I=\int_a^b x f(x) d x$ ....(i)
$I=\int_a^b(a+b-x) f(a+b-x) d x$ $\left[\because \int_a^b f(x) d x=\int_a^b f(a+b-x) d x\right]$
$\begin{array}{ll}\Rightarrow & I=\int_a^b(a+b-x) f(x) d x \\ \Rightarrow & I=(a+b) \int_a^b f(x) d x-I \quad \text { [Using Eq. (i)] }\end{array}$
$\begin{aligned} \Rightarrow & I+I & =(a+b) \int_a^b f(x) d x \\ \Rightarrow & 2 I & =(a+b) \int_a^b f(x) d x\end{aligned}$
$\Rightarrow \quad I=\left(\frac{a+b}{2}\right) \int_a^b f(x) d x$

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