Question

If $f(a+b-x)=f(x)$ then $\underset{a}{\overset{b}{\int }}x\cdot f(x)dx$ is equal to _______.

• A. $\frac{a+b}{2}\underset{a}{\overset{b}{\int }}f(x)dx$
• B. $\frac{a+b}{2}\underset{a}{\overset{b}{\int }}f(b+x)dx$
• C. $\frac{a+b}{2}\underset{a}{\overset{b}{\int }}f(b-x)dx$
• D. $\frac{b-a}{2}\underset{a}{\overset{b}{\int }}f(x)dx$

Answer 1

Answer: (A)

$I=\underset{a}{\overset{b}{\int }}x\cdot f(x)dx$
$=\int (a+b-x)f(a+b-x)=I$
add both
$\Rightarrow 2I=\underset{a}{\overset{b}{\int }}(a+b)f(x)dx$
$=\frac{a+b}{2}\underset{a}{\overset{b}{\int }}f(x)dx$