Question

If f (a + b –x) = f (x) then $\int _ { a } ^ { b } x$f (x) dx is equal to

• A. $\frac { a - b } { 2 }$ $\int _ { a } ^ { b } f ( x )$d x
• B. $\left( \frac { a + b } { 2 } \right)$ $\int _ { a } ^ { b } f ( x )$d x
• C. 0
• D. None of these

Answer 1

Answer: (B)

\left( \frac { a + b } { 2 } \right)$$\int _ { a } ^ { b } f ( x )d x

Answer 2

Let I = $\int _ { a } ^ { b }$(a+b-x) f (a +b-x) dx

= (a + b) $\int _ { a } ^ { b } f$ (a +b –x)dx - $\int _ { a } ^ { b }$x f (a + b – x) dx

= (a + b) $\int _ { a } ^ { b }$f (x) dx - $\int _ { a } ^ { b }$x f (x) dx.

Hence I = $\left( \frac { a + b } { 2 } \right)$ $\int _ { a } ^ { b }$f(x) dx.