Question

If f (a + b − x) = f (x), then \[\int\limits_a^b\] x f (x) dx is equal to

Options

• \[\frac{a + b}{2} \int\limits_a^b f\left( b - x \right) dx\]

   

• \[\frac{a + b}{2} \int\limits_a^b f\left( b + x \right) dx\]

   

• \[\frac{b - a}{2} \int\limits_a^b f\left( x \right) dx\]
• \[\frac{b + a}{2} \int\limits_a^b f\left( x \right) dx\]

Answer 1

\[\frac{a + b}{2} \int\limits_a^b f\left( x \right) dx\]

\[Let\, I = \int_a^b x f\left( x \right) d x .............(1)\]

\[ = \int_a^b \left( a + b - x \right) f\left( a + b - x \right) d x\]

\[ = \int_a^b \left( a + b - x \right) f\left( x \right) dx ...............(2)\]

\[ \text{Adding (1) and (2)}\]

\[2I = \int_a^b \left( x + a + b - x \right) f\left( x \right) d x\]

\[ = \left( a + b \right) \int_a^b f\left( x \right) d x \]

\[Hence\ I = \frac{a + b}{2} \int_a^b f\left( x \right) d x\]