If f(a+b-x)=f(x) then _a^b xf(x),dx is equal to:

Question

If $f(a+b-x)=f(x)$ then $\displaystyle\int_a^b xf(x)\,dx$ is equal to:

Accepted Answer

$$I = \int\limits_a^b {xf\left( x \right)} dx$$

$$ = \int\limits_a^b {\left( {a + b - x} \right)} f\left( {a + b - x} \right)dx$$

$$ = \left( {a + b} \right)\int\limits_a^b {f\left( {a + b - x} \right)} dx - \int\limits_a^b {xf} \left( {a + b - x} \right)dx$$

$$ = \left( {a + b} \right)\int\limits_a^b {f\left( x \right)dx}  - \int\limits_a^b {xf\left( x \right)dx} $$

$$\left[ {} \right.$$ As given that $$f\left( {a + b - x} \right) = f\left( x \right)$$ $$\left. {} \right]$$

$$2I = \left( {a + b} \right)\int\limits_a^b {f\left( x \right)} dx$$

$$ \Rightarrow I = {{\left( {a + b} \right)} \over 2}\int\limits_a^b {f\left( x \right)} dx$$

Solutions · 1