Question

$f ( x ) = f ( 2 - x )$ then $\int _ { 0.5 } ^ { 1.5 } x f ( x ) d x$ equals

• A. $\int _ { 0 } ^ { 1 } f ( x ) d x$
• B. $\int _ { 0.5 } ^ { 1.5 } f ( x ) d x$
• C. $2 \int _ { 0.5 } ^ { 1.5 } f ( x ) d x$
• D. 0

Answer 1

Answer: (B)

$\int _ { 0.5 } ^ { 1.5 } f ( x ) d x$

Answer 2

$I = \int _ { 0.5 } ^ { 1.5 } x f ( x ) d x = \int _ { 0.5 } ^ { 1.5 } ( 2 - x ) f ( 2 - x ) d x$ ,

$\left[ \because \int _ { a } ^ { b } f ( x ) d x = \int _ { a } ^ { b } f ( a + b - x ) d x \right]$

$= \int _ { 0.5 } ^ { 1.5 } ( 2 - x ) f ( x ) d x = 2 \int _ { 0.5 } ^ { 1.5 } f ( x ) d x - I$⇒$I = \int _ { 0.5 } ^ { 1.5 } f ( x ) d x$.