Question

aIf $f(a + b - x) = f(x)$ then $\int_{a}^{b}{xf(x)dx}$ is equal to

• A. $\frac{a + b}{2}\int_{a}^{b}{f(b - x)dx}$
• B. $\frac{a + b}{2}\int_{a}^{b}{f(x)dx}$
• C. $\frac{b - a}{2}\int_{a}^{b}{f(x)dx}$
• D. None of these

Answer 1

Answer: (B)

$\frac{a + b}{2}\int_{a}^{b}{f(x)dx}$

Answer 2

$I = \int_{a}^{b}{xf(x)dx}$ and $I = \int_{a}^{b}{(a + b - x)f(a + b - x)dx}$

⇒ $I = \int_{a}^{b}{(a + b - x)f(x)dx}$ ⇒ $I = \int_{a}^{b}{(a + b)f(x)dx - \int_{a}^{b}{xf(x)dx}}$

⇒ $2I = \left\lbrack \int_{a}^{b}{f(x)dx} \right\rbrack(a + b)$ ⇒ $I = \frac{a + b}{2}\int_{a}^{b}{f(x)dx}$

(8) $\int_{0}^{a}{xf(x)dx} = \frac{1}{2}a\int_{0}^{a}{f(x)dx}$ if $f(a - x) = f(x)$