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 = $\int_{a}^{b}x$ f (x) dx =$\int_{a}^{b}{}$ (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